An arithmetic sequence a_1, a_2, a_3, \dots has a first term a_1 = 4 and a common differen… — GMAT

An arithmetic sequence $a_1, a_2, a_3, \dots$ has a first term $a_1 = 4$ and a common difference $d = 3$ . A geometric sequence $b_1, b_2, b_3, \dots$ has a first term $b_1 = 2$ and a common ratio $r = 2$ . If the $k$ -th term of the arithmetic sequence and the $m$ -th term of the geometric sequence are both equal to $64$ , what is the value of $k + m$ ?

Answer: 27

Answer

The value of

k + m

is 27.

For the arithmetic sequence, the

k

-th term is

a_k = a_1 + (k-1)d

. Setting

4 + 3(k-1) = 64

gives

3(k-1) = 60

, so

k-1 = 20

and

k = 21

. For the geometric sequence, the

m

-th term is

b_m = b_1 r^{m-1}

. Setting

2 \cdot 2^{m-1} = 64

gives

2^m = 64

, which implies

m = 6

. Adding the two values gives

k + m = 21 + 6 = 27

Step-by-Step Solution

Determine the term position

k

in the arithmetic sequence.

k = 21

Using

a_k = a_1 + (k-1)d

, set

4 + 3(k-1) = 64 \implies 3(k-1) = 60 \implies k - 1 = 20 \implies k = 21

Determine the term position

m

in the geometric sequence.

m = 6

Using

b_m = b_1 \cdot r^{m-1}

, set

2 \cdot 2^{m-1} = 64 \implies 2^m = 64 \implies m = 6

Compute the sum of the two position indices

k

and

m

27

k + m = 21 + 6 = 27

Key Concept

Calculating term indices in arithmetic and geometric sequences using general term formulas

Estimated Time:1m 30s

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