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 solve this problem, we need to first understand the relationship between an Arithmetic Progression (A.P.) and a Geometric Progression (G.P.). Let's dissect the given data step-by-step:

1. The question states that the 2nd, 8th, and 44th terms of a non-constant A.P. form the 1st, 2nd, and 3rd terms of a G.P. respectively.
2. The formula for the \(n\)-th term of an A.P. is given by: \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference.
3. We know the first term of the A.P. is 1, thus \(a = 1\).
4. Let's determine the terms of the A.P.:
   • 2nd term: \(a_2 = a + d = 1 + d\)
   • 8th term: \(a_8 = a + 7d = 1 + 7d\)
   • 44th term: \(a_{44} = a + 43d = 1 + 43d\)
5. These terms form a G.P., so we have:
   • \(1 + 7d \div 1 + d = 1 + 43d \div 1 + 7d\)
6. Cross-multiplying gives:
   • \((1 + 7d)^2 = (1 + d)(1 + 43d)\)
   • Expanding both sides:
     • Left: \(1 + 14d + 49d^2\)
     • Right: \(1 + 43d + d + 43d^2 = 1 + 44d + 43d^2\)
   • We equate the expanded forms:
     • \(1 + 14d + 49d^2 = 1 + 44d + 43d^2\)
     • Simplifying: \(49d^2 + 14d = 43d^2 + 44d\)
     • This gives: \(6d^2 - 30d = 0\)
     • Factoring: \(6d(d - 5) = 0\)
7. From this, we get \(d = 0 \, \text{or} \, d = 5\). Since the A.P. is non-constant, \(d\) cannot be 0, thus \(d = 5\).
8. Now, we calculate the sum of the first 20 terms of the A.P. The formula for the sum of the first \(n\) terms is:
   • \(S_n = \frac{n}{2} [2a + (n-1)d]\)
   • Here, \(n = 20\), \(a = 1\), and \(d = 5\).
   • Plugging in the values: \(S_{20} = \frac{20}{2}(2 \times 1 + 19 \times 5)\)
   • Simplifying:
     • \(S_{20} = 10 \times (2 + 95) = 10 \times 97 = 970\)

Thus, the sum of the first 20 terms of the A.P. is 970.

Answer 2

Let the A.P. have the first term \( a = 1 \) and common difference \( d \). Then:

\[ \text{2nd term} = 1 + d, \quad \text{8th term} = 1 + 7d, \quad \text{44th term} = 1 + 43d \]

These terms are in G.P., so:

\[ (1 + 7d)^2 = (1 + d)(1 + 43d) \]

Expanding and simplifying:

\[ 1 + 49d^2 + 14d = 1 + 44d + 43d^2 \] \[ 6d^2 - 30d = 0 \] \[ d = 5 \]

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

\[ S_{20} = \frac{20}{2} \left[ 2 \cdot 1 + (20 - 1) \cdot 5 \right] \] \[ = 10 \cdot (2 + 95) = 10 \cdot 97 = 970 \]

Thus, the answer is:

970