Examkin

Question

Difficulty: MediumArithmetic and Geometric Progressions (AP and GP)

Arrange the following values related to arithmetic and geometric progressions in ascending order (from smallest to largest):

  1. 1The common ratio rr of a geometric progression with positive terms whose 2nd2^{\text{nd}} term is 55 and 4th4^{\text{th}} term is 8080
  2. 2The common difference dd of an arithmetic progression whose 1st1^{\text{st}} term is 33 and 5th5^{\text{th}} term is 3535
  3. 3The 4th4^{\text{th}} term of an arithmetic progression with first term 22 and common difference 33
  4. 4The sum of the first 33 terms of a geometric progression with first term 11 and common ratio 33

Answer

The correct order from smallest to largest value is: the common ratio of the GP (4), the common difference of the AP (8), the 4th term of the AP (11), and the sum of the first 3 terms of the GP (13).
Evaluating each progression property gives numerical values of 4, 8, 11, and 13 respectively. Ordering these from least to greatest results in the order: common ratio of the GP (4), common difference of the AP (8), 4th term of the AP (11), and sum of the first 3 terms of the GP (13).

Step-by-Step Solution

1
Calculate the value for the first item (common ratio rr)
Using T4=T2⋅r2T_4 = T_2 \cdot r^2, we have 80=5r2  ⟹  r2=16  ⟹  r=480 = 5 r^2 \implies r^2 = 16 \implies r = 4.
The terms of a GP follow Tn=arn−1T_n = a r^{n-1}.
2
Calculate the value for the second item (common difference dd)
Using T5=a+4dT_5 = a + 4d, we get 35=3+4d  ⟹  4d=32  ⟹  d=835 = 3 + 4d \implies 4d = 32 \implies d = 8.
The nth term of an AP is given by Tn=a+(n−1)dT_n = a + (n-1)d.
3
Calculate the value for the third item (4th4^{\text{th}} term of AP)
T4=2+(4−1)(3)=2+9=11T_4 = 2 + (4 - 1)(3) = 2 + 9 = 11.
Direct application of the AP nth term formula.
4
Calculate the value for the fourth item (Sum of first 33 terms of GP)
S3=1+3+9=13S_3 = 1 + 3 + 9 = 13 (or using Sn=a(rn−1)r−1=1(33−1)3−1=13S_n = \frac{a(r^n - 1)}{r - 1} = \frac{1(3^3 - 1)}{3 - 1} = 13).
Sum of a finite geometric sequence.
5
Compare and arrange the calculated numerical values in ascending order
4<8<11<134 < 8 < 11 < 13.
Ordering the numbers establishes the correct item sequence.

Key Concept

Evaluation of terms, differences, ratios, and sums in Arithmetic and Geometric Progressions.
Rate this question

Topics

Arithmetic and Geometric Progressions (AP and GP)

More questions from this topic