The 3^{\text{rd}} term of a geometric progression is 18 and the 6^{\text{th}} term is 486.… — JAMB UTME

The $3^{\text{rd}}$ term of a geometric progression is $18$ and the $6^{\text{th}}$ term is $486$ . What is the sum of the first $5$ terms of the progression?

$242$ Cevap
B
$162$
C
$80$
D
$728$

Cevap

The sum of the first

5

terms is

242

Using

T_n = a r^{n-1}

, we set up

a r^2 = 18

and

a r^5 = 486

. Dividing these gives

r^3 = 27

, so

r = 3

, which leads to

a = 2

. Applying

S_5 = \frac{a(r^5 - 1)}{r - 1}

yields

\frac{2(243 - 1)}{2} = 242

Adım Adım Çözüm

Express the given terms using the geometric progression

n^{\text{th}}

term formula

T_n = a r^{n-1}

T_3 = a r^2 = 18

and

T_6 = a r^5 = 486

This establishes a system of equations in terms of the first term

a

and common ratio

r

Divide the expression for

T_6

T_3

to determine

r

\frac{a r^5}{a r^2} = \frac{486}{18} \implies r^3 = 27 \implies r = 3.

Dividing the equations eliminates

a

and allows direct solution for the common ratio

r

Substitute

r = 3

back into

T_3 = a r^2 = 18

to solve for

a

a (3)^2 = 18 \implies 9a = 18 \implies a = 2.

Determining the first term

a

is required to evaluate the sum.

Calculate the sum of the first

5

terms using

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

S_5 = \frac{2(3^5 - 1)}{3 - 1} = \frac{2(243 - 1)}{2} = 242.

Applying the GP sum formula yields the required value.

Anahtar Kavram

Geometric Progression nth term and sum formulas

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