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Difficulty: HardArithmetic and Geometric Progressions (AP and GP)

The sum of the first nn terms of a sequence is given by Sn=2n2+3nS_n = 2n^2 + 3n. Find the 5th5^{\text{th}} term of a geometric progression whose first term is the 3rd3^{\text{rd}} term of this sequence, and whose common ratio is equal to the common difference of this sequence.

  1. 33283328Answer
  2. B
    1331213312
  3. C
    69126912
  4. D
    12801280

Answer

The 5th term of the geometric progression is 3328.
Evaluating S1S_1, S2S_2, and S3S_3 gives sequence terms T1=5T_1 = 5, T2=9T_2 = 9, and T3=13T_3 = 13. The common difference is d=9−5=4d = 9 - 5 = 4. Using T3=13T_3 = 13 as the first term of the GP and r=4r = 4 as the common ratio, the 5th term of the GP is 13×45−1=13×256=332813 \times 4^{5-1} = 13 \times 256 = 3328.

Step-by-Step Solution

1
Calculate the first few terms of the sequence using the sum formula Sn=2n2+3nS_n = 2n^2 + 3n.
S1=2(1)2+3(1)=5S_1 = 2(1)^2 + 3(1) = 5, S2=2(2)2+3(2)=14S_2 = 2(2)^2 + 3(2) = 14, S3=2(3)2+3(3)=27S_3 = 2(3)^2 + 3(3) = 27.
The sum formula gives cumulative sums, from which individual terms can be derived.
2
Find the 3rd term (T3T_3) and the common difference (dd) of the arithmetic progression.
T1=5T_1 = 5, T2=S2−S1=9T_2 = S_2 - S_1 = 9, T3=S3−S2=13T_3 = S_3 - S_2 = 13. Common difference d=T2−T1=4d = T_2 - T_1 = 4.
The difference between consecutive cumulative sums gives the individual sequence terms, and their constant difference gives the common difference.
3
Define the parameters of the geometric progression (GP).
First term of GP a=T3=13a = T_3 = 13, common ratio r=d=4r = d = 4.
The problem specifies that the first term of the GP is the 3rd term of the sequence and the common ratio equals the common difference.
4
Compute the 5th term of the geometric progression using Gn=a⋅rn−1G_n = a \cdot r^{n-1}.
G5=13⋅45−1=13⋅44=13⋅256=3328G_5 = 13 \cdot 4^{5-1} = 13 \cdot 4^4 = 13 \cdot 256 = 3328.
Applying the standard nth term formula for a GP with n=5n = 5.

Key Concept

Arithmetic Progression sum formula to term conversion and Geometric Progression nth term evaluation
Estimated Time:2m 0s
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Topics

Arithmetic and Geometric Progressions (AP and GP)

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