An arithmetic progression (A.P.) and a geometric progression (G.P.) both have a first term… — JAMB UTME

An arithmetic progression (A.P.) and a geometric progression (G.P.) both have a first term of $2$ . The common difference of the A.P. is $4$ . If the $5^{\text{th}}$ term of the A.P. is equal to the $3^{\text{rd}}$ term of the G.P., and the common ratio of the G.P. is positive, what is the $4^{\text{th}}$ term of the G.P.?

54Answer
B
162
C
18
D
14

Answer

First, evaluate the 5th term of the A.P. using

T_5 = 2 + (5-1)4 = 18

. Next, set the 3rd term of the G.P. equal to 18:

2 r^2 = 18

, giving

r^2 = 9

and

r = 3

. Finally, find the 4th term of the G.P. using

T_4 = 2 \times 3^3 = 54

Step-by-Step Solution

Calculate the 5th term of the A.P.

T_5^{(A.P.)} = a + (5-1)d = 2 + 4(4) = 18

The formula for the

n^{\text{th}}

term of an A.P. is

T_n = a + (n-1)d

Find the common ratio

r

of the G.P. by equating the 3rd term of the G.P. to 18

T_3^{(G.P.)} = a r^{3-1} = 2 r^2 = 18 \implies r^2 = 9 \implies r = 3

(since

r > 0

)

The formula for the

n^{\text{th}}

term of a G.P. is

T_n = a r^{n-1}

Compute the 4th term of the G.P.

T_4^{(G.P.)} = a r^{4-1} = 2 \times 3^3 = 2 \times 27 = 54

Substitute

a = 2

r = 3

, and

n = 4

into

T_n = a r^{n-1}

Key Concept

Connecting terms of Arithmetic and Geometric Progressions using their

n^{\text{th}}

term formulas

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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