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Difficulty: MediumArithmetic and Geometric Progressions (AP and GP)

An arithmetic progression (AP) has a first term of 22 and a common difference of 33. A geometric progression (GP) has a first term of 11 and a common ratio of 22. Arrange the following quantities in ascending order of their numerical values:

  1. 1The 4th4^{\text{th}} term of the GP
  2. 2The 4th4^{\text{th}} term of the AP
  3. 3The sum of the first 33 terms of the AP
  4. 4The 5th5^{\text{th}} term of the GP

Answer

The correct ascending order is the 4th term of the GP (8), followed by the 4th term of the AP (11), the sum of the first 3 terms of the AP (15), and the 5th term of the GP (16).
Evaluating each term individually gives: the 4th term of the GP equals 8, the 4th term of the AP equals 11, the sum of the first 3 terms of the AP equals 15, and the 5th term of the GP equals 16. Arranging these calculated values from smallest to largest gives the order 8, 11, 15, 16.

Step-by-Step Solution

1
Calculate the 4th term of the AP
T4=2+(4−1)×3=2+9=11T_4 = 2 + (4 - 1) \times 3 = 2 + 9 = 11
Using the AP nthn^{\text{th}} term formula Tn=a+(n−1)dT_n = a + (n-1)d.
2
Calculate the 5th term of the GP
T5=1×25−1=24=16T_5 = 1 \times 2^{5-1} = 2^4 = 16
Using the GP nthn^{\text{th}} term formula Tn=arn−1T_n = a r^{n-1}.
3
Calculate the sum of the first 3 terms of the AP
S3=32[2(2)+(3−1)3]=32[4+6]=15S_3 = \frac{3}{2}[2(2) + (3-1)3] = \frac{3}{2}[4 + 6] = 15
Using the AP sum formula Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}[2a + (n-1)d].
4
Calculate the 4th term of the GP
T4=1×24−1=23=8T_4 = 1 \times 2^{4-1} = 2^3 = 8
Using the GP nthn^{\text{th}} term formula Tn=arn−1T_n = a r^{n-1}.
5
Compare and arrange the values in ascending order
8<11<15<168 < 11 < 15 < 16
Arranging from smallest to largest numerical value.

Key Concept

Nth term and sum formulas of Arithmetic and Geometric Progressions
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Topics

Arithmetic and Geometric Progressions (AP and GP)

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