The 2^{\text{nd}} term of a geometric progression (GP) is 6 and the 5^{\text{th}} term is … — JAMB UTME

The $2^{\text{nd}}$ term of a geometric progression (GP) is $6$ and the $5^{\text{th}}$ term is $48$ . What is the sum of the first $6$ terms of the progression?

$189$ Cevap
B
$93$
C
$162$
D
$192$

Cevap

The sum of the first 6 terms of the geometric progression is 189.

By using the GP term formula

T_n = a r^{n-1}

, we establish

a r = 6

and

a r^4 = 48

. Dividing the fifth term by the second term yields

r^3 = 8

, giving

r = 2

. Substituting

r = 2

into

a r = 6

gives

a = 3

. Finally, using the sum formula

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

, we evaluate

S_6 = \frac{3(2^6 - 1)}{2 - 1} = 3(63) = 189

Adım Adım Çözüm

Set up equations for the given terms using the nth term formula

T_n = a r^{n-1}

T_2 = a r = 6

and

T_5 = a r^4 = 48

The nth term of a GP is defined by

T_n = a r^{n-1}

Solve for the common ratio

r

by dividing

T_5

T_2

\frac{a r^4}{a r} = \frac{48}{6} \implies r^3 = 8 \implies r = 2

Dividing the terms eliminates the first term

a

Find the first term

a

a(2) = 6 \implies a = 3

Substitute

r = 2

back into the equation for

T_2

Calculate the sum of the first 6 terms using

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

S_6 = \frac{3(2^6 - 1)}{2 - 1} = 3(64 - 1) = 3 \times 63 = 189

Apply the sum formula for a GP with

r > 1

Anahtar Kavram

Geometric Progression nth term and sum formulas

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