The first term of an arithmetic sequence is \frac{1}{2}, and the third term of the sequenc… — ACT

The first term of an arithmetic sequence is $\frac{1}{2}$ , and the third term of the sequence is $\frac{5}{6}$ . What is the sum of the first 6 terms of this sequence?

A
$\frac{9}{4}$
B
$3$
C
$5$
$\frac{11}{2}$ Answer
E
$\frac{21}{2}$

Answer

The sum of the first 6 terms is

\frac{11}{2}

To find the sum of the first 6 terms, we first determine the common difference

d

from the given terms:

a_3 = a_1 + 2d \Rightarrow \frac{5}{6} = \frac{1}{2} + 2d \Rightarrow 2d = \frac{1}{3} \Rightarrow d = \frac{1}{6}

. Then, we apply the arithmetic series sum formula:

S_6 = \frac{6}{2}[2(\frac{1}{2}) + 5(\frac{1}{6})] = 3[1 + \frac{5}{6}] = 3(\frac{11}{6}) = \frac{11}{2}

Step-by-Step Solution

Find the common difference

d

using the formula for the

n

-th term of an arithmetic sequence

a_n = a_1 + (n - 1)d

with the given values

a_1 = \frac{1}{2}

and

a_3 = \frac{5}{6}

2d = \frac{5}{6} - \frac{1}{2} \Rightarrow 2d = \frac{1}{3} \Rightarrow d = \frac{1}{6}

We need to find the common difference to determine the subsequent terms and calculate the sum of the sequence.

Use the sum formula

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

with

n = 6

a_1 = \frac{1}{2}

, and

d = \frac{1}{6}

to find the sum of the first 6 terms.

S_6 = \frac{6}{2}[2(\frac{1}{2}) + (6 - 1)(\frac{1}{6})] = 3[1 + \frac{5}{6}] = 3(\frac{11}{6}) = \frac{11}{2}

This formula sums the first 6 terms of the arithmetic sequence directly.

Key Concept

Sum of the first

n

terms of an arithmetic sequence using fractional terms

Alternative Method

Alternatively, you can list the first 6 terms of the sequence and add them directly:

\frac{3}{6}, \frac{4}{6}, \frac{5}{6}, \frac{6}{6}, \frac{7}{6}, \frac{8}{6}

. Adding these gives

\frac{33}{6} = \frac{11}{2}

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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