← 2-2 · Sum a repeating number sequence · Sum of Evenly Spaced Numbers via the Middle

Sum a repeating number sequence · 10 practice problems

4.OA.C.53.OA.A.3

Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 answer: 46

The numbers below are arranged according to a rule. Find the repeating block, then find the sum of the numbers when they are written out to the $13$ th term.

$6,\ 1,\ 3,\ 6,\ 1,\ 3,\ 6,\ 1,\ \ldots$

Show solution

Understand

The list 6, 1, 3, 6, 1, 3, ... repeats a block. Find the repeating block, then add up the first 13 terms.

Givens

The sequence repeats the block 6, 1, 3.
The same block of numbers repeats over and over.

Unknowns

The sum of the first 13 terms of the sequence.

Constraints

The repeating block is 6, 1, 3 (length 3).
We need exactly the first 13 terms.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list repeats the block 6, 1, 3, so I find the block, count how many whole blocks fit into 13 terms, then add up the block sums (plus any leftover terms).

Execute

#5 Look for a Pattern 4.OA.C.5

The numbers repeat in groups of 3: 6, 1, 3. One block sums to 6 + 1 + 3 = 10.

6 + 1 + 3 = 10

Spotting that the same numbers come back tells you the block and its repeat length.

#9 Solve an Easier Related Problem 3.OA.A.3

Each block has 3 terms, and 13 terms divided into groups of 3 gives 4 complete blocks with 1 term left over.

13 \div 3 = 4 \text{ R } 1

Grouping 13 terms by 3 shows how many whole blocks fit.

#5 Look for a Pattern 3.OA.A.1

4 blocks, each summing to 10, give a total of 4 x 10 = 40.

4 \times 10 = 40

Repeated equal groups are quickest added by multiplying.

#5 Look for a Pattern 3.OA.A.3

After 4 whole blocks, the next 1 term are 6, which add 6. The grand total is 40 + 6 = 46.

40 + 6 = 46

A partial block at the end just adds its first few terms.

Answer: 46

Review

4 full blocks give 40, plus the 1 leftover term summing to 6, lands on 46.

You could write out all 13 terms and add them one by one to confirm the total reaches 46.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing the repeating block 6, 1, 3.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 13 terms into groups of 3 to count blocks.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Multiplying 4 blocks by the block sum 10.

💡 Find the repeating block, see how many fit, and multiply: 4 blocks of (6 + 1 + 3) is 4 x 10 = 40!

Variant 2 answer: 39

The numbers below are arranged according to a rule. Find the repeating block, then find the sum of the numbers when they are written out to the $11$ th term.

$5,\ 2,\ 4,\ 3,\ 5,\ 2,\ 4,\ 3,\ 5,\ 2,\ \ldots$

Show solution

Understand

The list 5, 2, 4, 3, 5, 2, 4, 3, ... repeats a block. Find the repeating block, then add up the first 11 terms.

Givens

The sequence repeats the block 5, 2, 4, 3.
The same block of numbers repeats over and over.

Unknowns

The sum of the first 11 terms of the sequence.

Constraints

The repeating block is 5, 2, 4, 3 (length 4).
We need exactly the first 11 terms.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list repeats the block 5, 2, 4, 3, so I find the block, count how many whole blocks fit into 11 terms, then add up the block sums (plus any leftover terms).

Execute

#5 Look for a Pattern 4.OA.C.5

The numbers repeat in groups of 4: 5, 2, 4, 3. One block sums to 5 + 2 + 4 + 3 = 14.

5 + 2 + 4 + 3 = 14

Spotting that the same numbers come back tells you the block and its repeat length.

#9 Solve an Easier Related Problem 3.OA.A.3

Each block has 4 terms, and 11 terms divided into groups of 4 gives 2 complete blocks with 3 terms left over.

11 \div 4 = 2 \text{ R } 3

Grouping 11 terms by 4 shows how many whole blocks fit.

#5 Look for a Pattern 3.OA.A.1

2 blocks, each summing to 14, give a total of 2 x 14 = 28.

2 \times 14 = 28

Repeated equal groups are quickest added by multiplying.

#5 Look for a Pattern 3.OA.A.3

After 2 whole blocks, the next 3 terms are 5 + 2 + 4, which add 11. The grand total is 28 + 11 = 39.

28 + 11 = 39

A partial block at the end just adds its first few terms.

Answer: 39

Review

2 full blocks give 28, plus the 3 leftover terms summing to 11, lands on 39.

You could write out all 11 terms and add them one by one to confirm the total reaches 39.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing the repeating block 5, 2, 4, 3.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 11 terms into groups of 4 to count blocks.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Multiplying 2 blocks by the block sum 14.

💡 Find the repeating block, see how many fit, and multiply: 2 blocks of (5 + 2 + 4 + 3) is 2 x 14 = 28!

Variant 3 answer: 58

The numbers below are arranged according to a rule. Find the repeating block, then find the sum of the numbers when they are written out to the $14$ th term.

$1,\ 9,\ 2,\ 1,\ 9,\ 2,\ 1,\ 9,\ \ldots$

Show solution

Understand

The list 1, 9, 2, 1, 9, 2, ... repeats a block. Find the repeating block, then add up the first 14 terms.

Givens

The sequence repeats the block 1, 9, 2.
The same block of numbers repeats over and over.

Unknowns

The sum of the first 14 terms of the sequence.

Constraints

The repeating block is 1, 9, 2 (length 3).
We need exactly the first 14 terms.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list repeats the block 1, 9, 2, so I find the block, count how many whole blocks fit into 14 terms, then add up the block sums (plus any leftover terms).

Execute

#5 Look for a Pattern 4.OA.C.5

The numbers repeat in groups of 3: 1, 9, 2. One block sums to 1 + 9 + 2 = 12.

1 + 9 + 2 = 12

Spotting that the same numbers come back tells you the block and its repeat length.

#9 Solve an Easier Related Problem 3.OA.A.3

Each block has 3 terms, and 14 terms divided into groups of 3 gives 4 complete blocks with 2 terms left over.

14 \div 3 = 4 \text{ R } 2

Grouping 14 terms by 3 shows how many whole blocks fit.

#5 Look for a Pattern 3.OA.A.1

4 blocks, each summing to 12, give a total of 4 x 12 = 48.

4 \times 12 = 48

Repeated equal groups are quickest added by multiplying.

#5 Look for a Pattern 3.OA.A.3

After 4 whole blocks, the next 2 terms are 1 + 9, which add 10. The grand total is 48 + 10 = 58.

48 + 10 = 58

A partial block at the end just adds its first few terms.

Answer: 58

Review

4 full blocks give 48, plus the 2 leftover terms summing to 10, lands on 58.

You could write out all 14 terms and add them one by one to confirm the total reaches 58.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing the repeating block 1, 9, 2.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 14 terms into groups of 3 to count blocks.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Multiplying 4 blocks by the block sum 12.

💡 Find the repeating block, see how many fit, and multiply: 4 blocks of (1 + 9 + 2) is 4 x 12 = 48!

Variant 4 answer: 60

The numbers below are arranged according to a rule. Find the repeating block, then find the sum of the numbers when they are written out to the $24$ th term.

$2,\ 2,\ 5,\ 1,\ 2,\ 2,\ 5,\ 1,\ 2,\ 2,\ \ldots$

Show solution

Understand

The list 2, 2, 5, 1, 2, 2, 5, 1, ... repeats a block. Find the repeating block, then add up the first 24 terms.

Givens

The sequence repeats the block 2, 2, 5, 1.
The same block of numbers repeats over and over.

Unknowns

The sum of the first 24 terms of the sequence.

Constraints

The repeating block is 2, 2, 5, 1 (length 4).
We need exactly the first 24 terms.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list repeats the block 2, 2, 5, 1, so I find the block, count how many whole blocks fit into 24 terms, then add up the block sums (plus any leftover terms).

Execute

#5 Look for a Pattern 4.OA.C.5

The numbers repeat in groups of 4: 2, 2, 5, 1. One block sums to 2 + 2 + 5 + 1 = 10.

2 + 2 + 5 + 1 = 10

Spotting that the same numbers come back tells you the block and its repeat length.

#9 Solve an Easier Related Problem 3.OA.A.3

Each block has 4 terms, and 24 terms divided into groups of 4 gives 6 complete blocks with 0 terms left over.

24 \div 4 = 6

Grouping 24 terms by 4 shows how many whole blocks fit.

#5 Look for a Pattern 3.OA.A.1

6 blocks, each summing to 10, give a total of 6 x 10 = 60.

6 \times 10 = 60

Repeated equal groups are quickest added by multiplying.

Answer: 60

Review

24 terms split evenly into 6 whole blocks with nothing left over, so 6 x 10 = 60 is exact.

You could write out all 24 terms and add them one by one to confirm the total reaches 60.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing the repeating block 2, 2, 5, 1.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 24 terms into groups of 4 to count blocks.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Multiplying 6 blocks by the block sum 10.

💡 Find the repeating block, see how many fit, and multiply: 6 blocks of (2 + 2 + 5 + 1) is 6 x 10 = 60!

Variant 5 answer: 62

The numbers below are arranged according to a rule. Find the repeating block, then find the sum of the numbers when they are written out to the $20$ th term.

$4,\ 4,\ 1,\ 4,\ 4,\ 1,\ 4,\ 4,\ \ldots$

Show solution

Understand

The list 4, 4, 1, 4, 4, 1, ... repeats a block. Find the repeating block, then add up the first 20 terms.

Givens

The sequence repeats the block 4, 4, 1.
The same block of numbers repeats over and over.

Unknowns

The sum of the first 20 terms of the sequence.

Constraints

The repeating block is 4, 4, 1 (length 3).
We need exactly the first 20 terms.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list repeats the block 4, 4, 1, so I find the block, count how many whole blocks fit into 20 terms, then add up the block sums (plus any leftover terms).

Execute

#5 Look for a Pattern 4.OA.C.5

The numbers repeat in groups of 3: 4, 4, 1. One block sums to 4 + 4 + 1 = 9.

4 + 4 + 1 = 9

Spotting that the same numbers come back tells you the block and its repeat length.

#9 Solve an Easier Related Problem 3.OA.A.3

Each block has 3 terms, and 20 terms divided into groups of 3 gives 6 complete blocks with 2 terms left over.

20 \div 3 = 6 \text{ R } 2

Grouping 20 terms by 3 shows how many whole blocks fit.

#5 Look for a Pattern 3.OA.A.1

6 blocks, each summing to 9, give a total of 6 x 9 = 54.

6 \times 9 = 54

Repeated equal groups are quickest added by multiplying.

#5 Look for a Pattern 3.OA.A.3

After 6 whole blocks, the next 2 terms are 4 + 4, which add 8. The grand total is 54 + 8 = 62.

54 + 8 = 62

A partial block at the end just adds its first few terms.

Answer: 62

Review

6 full blocks give 54, plus the 2 leftover terms summing to 8, lands on 62.

You could write out all 20 terms and add them one by one to confirm the total reaches 62.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing the repeating block 4, 4, 1.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 20 terms into groups of 3 to count blocks.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Multiplying 6 blocks by the block sum 9.

💡 Find the repeating block, see how many fit, and multiply: 6 blocks of (4 + 4 + 1) is 6 x 9 = 54!

Variant 6 answer: 48

The numbers below are arranged according to a rule. Find the repeating block, then find the sum of the numbers when they are written out to the $16$ th term.

$1,\ 6,\ 2,\ 3,\ 1,\ 6,\ 2,\ 3,\ 1,\ 6,\ \ldots$

Show solution

Understand

The list 1, 6, 2, 3, 1, 6, 2, 3, ... repeats a block. Find the repeating block, then add up the first 16 terms.

Givens

The sequence repeats the block 1, 6, 2, 3.
The same block of numbers repeats over and over.

Unknowns

The sum of the first 16 terms of the sequence.

Constraints

The repeating block is 1, 6, 2, 3 (length 4).
We need exactly the first 16 terms.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list repeats the block 1, 6, 2, 3, so I find the block, count how many whole blocks fit into 16 terms, then add up the block sums (plus any leftover terms).

Execute

#5 Look for a Pattern 4.OA.C.5

The numbers repeat in groups of 4: 1, 6, 2, 3. One block sums to 1 + 6 + 2 + 3 = 12.

1 + 6 + 2 + 3 = 12

Spotting that the same numbers come back tells you the block and its repeat length.

#9 Solve an Easier Related Problem 3.OA.A.3

Each block has 4 terms, and 16 terms divided into groups of 4 gives 4 complete blocks with 0 terms left over.

16 \div 4 = 4

Grouping 16 terms by 4 shows how many whole blocks fit.

#5 Look for a Pattern 3.OA.A.1

4 blocks, each summing to 12, give a total of 4 x 12 = 48.

4 \times 12 = 48

Repeated equal groups are quickest added by multiplying.

Answer: 48

Review

16 terms split evenly into 4 whole blocks with nothing left over, so 4 x 12 = 48 is exact.

You could write out all 16 terms and add them one by one to confirm the total reaches 48.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing the repeating block 1, 6, 2, 3.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 16 terms into groups of 4 to count blocks.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Multiplying 4 blocks by the block sum 12.

💡 Find the repeating block, see how many fit, and multiply: 4 blocks of (1 + 6 + 2 + 3) is 4 x 12 = 48!

Variant 7 answer: 90

The numbers below are arranged according to a rule. Find the repeating block, then find the sum of the numbers when they are written out to the $18$ th term.

$3,\ 7,\ 3,\ 7,\ 3,\ 7,\ \ldots$

Show solution

Understand

The list 3, 7, 3, 7, ... repeats a block. Find the repeating block, then add up the first 18 terms.

Givens

The sequence repeats the block 3, 7.
The same block of numbers repeats over and over.

Unknowns

The sum of the first 18 terms of the sequence.

Constraints

The repeating block is 3, 7 (length 2).
We need exactly the first 18 terms.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list repeats the block 3, 7, so I find the block, count how many whole blocks fit into 18 terms, then add up the block sums (plus any leftover terms).

Execute

#5 Look for a Pattern 4.OA.C.5

The numbers repeat in groups of 2: 3, 7. One block sums to 3 + 7 = 10.

3 + 7 = 10

Spotting that the same numbers come back tells you the block and its repeat length.

#9 Solve an Easier Related Problem 3.OA.A.3

Each block has 2 terms, and 18 terms divided into groups of 2 gives 9 complete blocks with 0 terms left over.

18 \div 2 = 9

Grouping 18 terms by 2 shows how many whole blocks fit.

#5 Look for a Pattern 3.OA.A.1

9 blocks, each summing to 10, give a total of 9 x 10 = 90.

9 \times 10 = 90

Repeated equal groups are quickest added by multiplying.

Answer: 90

Review

18 terms split evenly into 9 whole blocks with nothing left over, so 9 x 10 = 90 is exact.

You could write out all 18 terms and add them one by one to confirm the total reaches 90.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing the repeating block 3, 7.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 18 terms into groups of 2 to count blocks.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Multiplying 9 blocks by the block sum 10.

💡 Find the repeating block, see how many fit, and multiply: 9 blocks of (3 + 7) is 9 x 10 = 90!

Variant 8 answer: 63

The numbers below are arranged according to a rule. Find the repeating block, then find the sum of the numbers when they are written out to the $21$ th term.

$3,\ 3,\ 3,\ 3,\ 3,\ 3,\ 3,\ 3,\ \ldots$

Show solution

Understand

The list 3, 3, 3, 3, 3, 3, ... repeats a block. Find the repeating block, then add up the first 21 terms.

Givens

The sequence repeats the block 3, 3, 3.
The same block of numbers repeats over and over.

Unknowns

The sum of the first 21 terms of the sequence.

Constraints

The repeating block is 3, 3, 3 (length 3).
We need exactly the first 21 terms.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list repeats the block 3, 3, 3, so I find the block, count how many whole blocks fit into 21 terms, then add up the block sums (plus any leftover terms).

Execute

#5 Look for a Pattern 4.OA.C.5

The numbers repeat in groups of 3: 3, 3, 3. One block sums to 3 + 3 + 3 = 9.

3 + 3 + 3 = 9

Spotting that the same numbers come back tells you the block and its repeat length.

#9 Solve an Easier Related Problem 3.OA.A.3

Each block has 3 terms, and 21 terms divided into groups of 3 gives 7 complete blocks with 0 terms left over.

21 \div 3 = 7

Grouping 21 terms by 3 shows how many whole blocks fit.

#5 Look for a Pattern 3.OA.A.1

7 blocks, each summing to 9, give a total of 7 x 9 = 63.

7 \times 9 = 63

Repeated equal groups are quickest added by multiplying.

Answer: 63

Review

21 terms split evenly into 7 whole blocks with nothing left over, so 7 x 9 = 63 is exact.

You could write out all 21 terms and add them one by one to confirm the total reaches 63.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing the repeating block 3, 3, 3.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 21 terms into groups of 3 to count blocks.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Multiplying 7 blocks by the block sum 9.

💡 Find the repeating block, see how many fit, and multiply: 7 blocks of (3 + 3 + 3) is 7 x 9 = 63!

Variant 9 answer: 44

The numbers below are arranged according to a rule. Find the repeating block, then find the sum of the numbers when they are written out to the $12$ th term.

$2,\ 5,\ 4,\ 2,\ 5,\ 4,\ 2,\ 5,\ \ldots$

Show solution

Understand

The list 2, 5, 4, 2, 5, 4, ... repeats a block. Find the repeating block, then add up the first 12 terms.

Givens

The sequence repeats the block 2, 5, 4.
The same block of numbers repeats over and over.

Unknowns

The sum of the first 12 terms of the sequence.

Constraints

The repeating block is 2, 5, 4 (length 3).
We need exactly the first 12 terms.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list repeats the block 2, 5, 4, so I find the block, count how many whole blocks fit into 12 terms, then add up the block sums (plus any leftover terms).

Execute

#5 Look for a Pattern 4.OA.C.5

The numbers repeat in groups of 3: 2, 5, 4. One block sums to 2 + 5 + 4 = 11.

2 + 5 + 4 = 11

Spotting that the same numbers come back tells you the block and its repeat length.

#9 Solve an Easier Related Problem 3.OA.A.3

Each block has 3 terms, and 12 terms divided into groups of 3 gives 4 complete blocks with 0 terms left over.

12 \div 3 = 4

Grouping 12 terms by 3 shows how many whole blocks fit.

#5 Look for a Pattern 3.OA.A.1

4 blocks, each summing to 11, give a total of 4 x 11 = 44.

4 \times 11 = 44

Repeated equal groups are quickest added by multiplying.

Answer: 44

Review

12 terms split evenly into 4 whole blocks with nothing left over, so 4 x 11 = 44 is exact.

You could write out all 12 terms and add them one by one to confirm the total reaches 44.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing the repeating block 2, 5, 4.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 12 terms into groups of 3 to count blocks.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Multiplying 4 blocks by the block sum 11.

💡 Find the repeating block, see how many fit, and multiply: 4 blocks of (2 + 5 + 4) is 4 x 11 = 44!

Variant 10 answer: 60

The numbers below are arranged according to a rule. Find the repeating block, then find the sum of the numbers when they are written out to the $15$ th term.

$1,\ 3,\ 8,\ 1,\ 3,\ 8,\ 1,\ 3,\ \ldots$

Show solution

Understand

The list 1, 3, 8, 1, 3, 8, ... repeats a block. Find the repeating block, then add up the first 15 terms.

Givens

The sequence repeats the block 1, 3, 8.
The same block of numbers repeats over and over.

Unknowns

The sum of the first 15 terms of the sequence.

Constraints

The repeating block is 1, 3, 8 (length 3).
We need exactly the first 15 terms.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list repeats the block 1, 3, 8, so I find the block, count how many whole blocks fit into 15 terms, then add up the block sums (plus any leftover terms).

Execute

#5 Look for a Pattern 4.OA.C.5

The numbers repeat in groups of 3: 1, 3, 8. One block sums to 1 + 3 + 8 = 12.

1 + 3 + 8 = 12

Spotting that the same numbers come back tells you the block and its repeat length.

#9 Solve an Easier Related Problem 3.OA.A.3

Each block has 3 terms, and 15 terms divided into groups of 3 gives 5 complete blocks with 0 terms left over.

15 \div 3 = 5

Grouping 15 terms by 3 shows how many whole blocks fit.

#5 Look for a Pattern 3.OA.A.1

5 blocks, each summing to 12, give a total of 5 x 12 = 60.

5 \times 12 = 60

Repeated equal groups are quickest added by multiplying.

Answer: 60

Review

15 terms split evenly into 5 whole blocks with nothing left over, so 5 x 12 = 60 is exact.

You could write out all 15 terms and add them one by one to confirm the total reaches 60.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing the repeating block 1, 3, 8.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 15 terms into groups of 3 to count blocks.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Multiplying 5 blocks by the block sum 12.

💡 Find the repeating block, see how many fit, and multiply: 5 blocks of (1 + 3 + 8) is 5 x 12 = 60!