An arithmetic sequence a_1, a_2, a_3, \dots has a first term a_1 and a common difference d… — ACT

An arithmetic sequence $a_1, a_2, a_3, \dots$ has a first term $a_1$ and a common difference $d$ , where both $a_1$ and $d$ are non-zero. Let $S_n$ represent the sum of the first $n$ terms of this sequence. If the ratio $\frac{S_{3n}}{S_n}$ is equal to a constant value $C$ for all positive integers $n$ , what is the ratio of the tenth term, $a_{10}$ , to the first term, $a_1$ ?

A
10
B
21
19Cevap
D
9
E
46

Cevap

The ratio of the tenth term to the first term is 19.

The sum of the first

n

terms of an arithmetic sequence is

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

. For

n=1

, the ratio is

\frac{S_3}{S_1} = 3 + \frac{3d}{a_1}

. For

n=2

, the ratio is

\frac{S_6}{S_2} = \frac{6a_1 + 15d}{2a_1 + d}

. Setting these equal because the ratio is constant for all positive integers

n

gives

3 + \frac{3d}{a_1} = \frac{6a_1 + 15d}{2a_1 + d}

, which simplifies to

\frac{3a_1+3d}{a_1} = \frac{6a_1+15d}{2a_1+d}

. Cross-multiplying and simplifying yields

3d^2 = 6a_1 d

. Since

d \neq 0

, we have

d = 2a_1

. The tenth term is

a_{10} = a_1 + 9d = a_1 + 9(2a_1) = 19a_1

. Thus, the ratio of the tenth term to the first term is

19

Adım Adım Çözüm

Write the formula for the sum of the first

n

terms of an arithmetic sequence,

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

, and find the expressions for

S_1

and

S_3

S_1 = a_1

and

S_3 = 3a_1 + 3d

To evaluate the ratio

\frac{S_{3n}}{S_n}

for the case where

n = 1

Write the expressions for

S_2

and

S_6

using the arithmetic sum formula.

S_2 = 2a_1 + d

and

S_6 = 6a_1 + 15d

To evaluate the ratio

\frac{S_{3n}}{S_n}

for the case where

n = 2

Equate the ratio for

n = 1

to the ratio for

n = 2

since the ratio

\frac{S_{3n}}{S_n}

must be constant for all

n

\frac{3a_1 + 3d}{a_1} = \frac{6a_1 + 15d}{2a_1 + d}

To set up an algebraic equation to find the relationship between the first term

a_1

and the common difference

d

Solve the equation for

d

in terms of

a_1

by cross-multiplying and simplifying.

d = 2a_1

Cross-multiplying gives

(3a_1 + 3d)(2a_1 + d) = a_1(6a_1 + 15d)

, which expands to

6a_1^2 + 9a_1 d + 3d^2 = 6a_1^2 + 15a_1 d

. Subtracting

6a_1^2

and

9a_1 d

from both sides yields

3d^2 = 6a_1 d

. Since

d \neq 0

, dividing by

3d

gives

d = 2a_1

Substitute

d = 2a_1

into the formula for the tenth term,

a_{10} = a_1 + 9d

, and calculate the ratio

\frac{a_{10}}{a_1}

a_{10} = 19a_1

, so the ratio is

19

To determine the final value of the requested ratio.

Anahtar Kavram

Relating arithmetic sequence term and sum formulas through systems of algebraic equations.

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