The sum of the first 4 terms of a geometric progression (GP) with a common ratio of 2 is 4… — JAMB UTME

The sum of the first $4$ terms of a geometric progression (GP) with a common ratio of $2$ is $45$ . What is the $6^{\text{th}}$ term of the progression?

Cevap: 96

Cevap

The

6^{\text{th}}

term of the geometric progression is

96

Using the sum formula

S_4 = \frac{a(2^4 - 1)}{2 - 1} = 45

gives

15a = 45

, so the first term

a

3

. Substituting

a = 3

and

r = 2

into the term formula

T_6 = a r^5

gives

3 \times 32 = 96

Adım Adım Çözüm

Express the sum of the first 4 terms using the GP sum formula to find the first term

a

Setting up

45 = \frac{a(2^4 - 1)}{2 - 1}

yields

15a = 45

, so

a = 3

The sum formula

S_n = \frac{a(r^n - 1)}{r - 1}

allows us to isolate the unknown initial term

a

when

S_4

and

r

are given.

Calculate the

6^{\text{th}}

term

T_6

using the

n^{\text{th}}

term formula

T_n = a r^{n-1}

T_6 = 3 \times 2^{6-1} = 3 \times 32 = 96

The exponent for the common ratio in the

n^{\text{th}}

term formula is

n - 1

, giving

5

as the exponent.

Anahtar Kavram

Sum and

n^{\text{th}}

term of a Geometric Progression

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