In an arithmetic sequence, the sum of the first 4 terms is 28 and the sum of the first 8 t… — GMAT

In an arithmetic sequence, the sum of the first $4$ terms is $28$ and the sum of the first $8$ terms is $88$ . What is the $10\text{th}$ term of this sequence?

A
$18$
B
$20$
$22$ Answer
D
$24$
E
$26$

Answer

The 10th term of the sequence is 22.

Using the arithmetic series sum formula

S_n = \frac{n}{2}(2a_1 + (n-1)d)

, the conditions yield two linear equations:

2a_1 + 3d = 14

and

2a_1 + 7d = 22

. Subtracting these equations gives

4d = 8

, so

d = 2

. Substituting

d = 2

back into

2a_1 + 3(2) = 14

yields

a_1 = 4

. The 10th term is then calculated as

a_1 + 9d = 4 + 9(2) = 22

Step-by-Step Solution

Express the given sums using the arithmetic series sum formula

S_n = \frac{n}{2}(2a_1 + (n-1)d)

For

S_4 = 28

\frac{4}{2}(2a_1 + 3d) = 28 \implies 2a_1 + 3d = 14

.
For

S_8 = 88

\frac{8}{2}(2a_1 + 7d) = 88 \implies 2a_1 + 7d = 22

Setting up linear equations in terms of the first term

a_1

and common difference

d

allows us to solve for both sequence parameters.

Subtract the first equation from the second equation to solve for

d

(2a_1 + 7d) - (2a_1 + 3d) = 22 - 14 \implies 4d = 8 \implies d = 2

Eliminating

a_1

isolates the common difference

d

Substitute

d = 2

back into the first equation to solve for

a_1

2a_1 + 3(2) = 14 \implies 2a_1 + 6 = 14 \implies 2a_1 = 8 \implies a_1 = 4

Finding

a_1

completes the essential parameters of the sequence.

Calculate the 10th term using the formula

a_n = a_1 + (n-1)d

a_{10} = 4 + (10 - 1)(2) = 4 + 18 = 22

Evaluating the formula at

n = 10

provides the target term.

Key Concept

Arithmetic sequence term and series sum formulas.

Estimated Time:2m 0s

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