Question

Let 2nd, 8th, and 44th terms of a non-constant A.P. be respectively the 1st, 2nd, and 3rd terms of a G.P. If the first term of A.P. is 1, then the sum of the first 20 terms is equal to

• A. 980
• B. 960
• C. 990
• D. 970

Answer 1

The Correct Option is D

To address this, we must first establish the relationship between Arithmetic Progressions (A.P.) and Geometric Progressions (G.P.). The provided data is analyzed as follows:

1. The problem states that specific terms of a non-constant A.P. correspond to the initial terms of a G.P.
2. The general formula for the \(n\)-th term of an A.P. is \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference.
3. The first term of the A.P. is given as \(a = 1\).
4. The relevant terms of the A.P. are calculated:
   • 2nd term: \(a_2 = 1 + d\)
   • 8th term: \(a_8 = 1 + 7d\)
   • 44th term: \(a_{44} = 1 + 43d\)
5. These terms form a G.P., implying a constant ratio:
   • \(\frac{1 + 7d}{1 + d} = \frac{1 + 43d}{1 + 7d}\)
6. Cross-multiplication yields:
   • \((1 + 7d)^2 = (1 + d)(1 + 43d)\)
   • Expanding both sides:
     • Left side: \(1 + 14d + 49d^2\)
     • Right side: \(1 + 44d + 43d^2\)
   • Equating the expanded expressions:
     • \(1 + 14d + 49d^2 = 1 + 44d + 43d^2\)
     • Simplification leads to: \(6d^2 - 30d = 0\)
     • Factoring out \(6d\): \(6d(d - 5) = 0\)
7. The possible values for \(d\) are \(d = 0\) or \(d = 5\). Since the A.P. is non-constant, \(d eq 0\), therefore \(d = 5\).
8. The sum of the first 20 terms of the A.P. is calculated using the formula \(S_n = \frac{n}{2} [2a + (n-1)d]\).
   • With \(n = 20\), \(a = 1\), and \(d = 5\):
   • \(S_{20} = \frac{20}{2}(2 \times 1 + 19 \times 5)\)
   • Calculation:
     • \(S_{20} = 10 \times (2 + 95) = 10 \times 97 = 970\)

The sum of the first 20 terms of the A.P. is 970.