The second term of a geometric progression (G.P.) is 6 and its fifth term is 48. What is t… — JAMB UTME

The second term of a geometric progression (G.P.) is $6$ and its fifth term is $48$ . What is the sum of the first $6$ terms of the progression?

189

Using the geometric progression terms

T_2 = a r = 6

and

T_5 = a r^4 = 48

, dividing

T_5

T_2

gives

r^3 = 8

, so

r = 2

. Substituting

r = 2

into

a r = 6

yields

a = 3

. The sum of the first 6 terms is calculated as

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

Set up equations using the

n^{\text{th}}

term formula for a G.P.,

T_n = a r^{n-1}

T_2 = a r = 6

and

T_5 = a r^4 = 48

Relate given terms to the first term

a

and common ratio

r

Divide the equation for

T_5

by the equation for

T_2

to find the common ratio

r

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

Eliminate the variable

a

to solve for

r

Substitute

r = 2

back into

a r = 6

to find the first term

a

a(2) = 6 \implies a = 3

Determine the first term of the progression.

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(63) = 189

Apply the sum formula for a finite geometric progression.

Geometric Progression (G.P.)

n^{\text{th}}

term and sum of finite terms

Question