Question

Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is:

• A. 122
• B. 84
• C. 90
• D. 108

Hint:

For problems involving arithmetic progressions, use the formulas for the sum of terms and the nth term to solve for unknowns. You can often find the common difference by using conditions for the sum.

Answer 1

The Correct Option is D

Step 1: Define the first term of the arithmetic progression (A.P.) as \( a \) and the common difference as \( d \). The sum of the initial three terms is expressed as: \[ S_3 = 3a + 3d = 54 \quad \Rightarrow \quad a + d = 18 \] From this, we derive \( a = 18 - d \).
Step 2: The sum of the first 20 terms is formulated as: \[ S_{20} = \frac{20}{2} \times (2a + 19d) \] Determine the values of \( a \) and \( d \) that ensure the sum falls within the range of 1600 to 1800. 
Step 3: Once \( a \) and \( d \) are determined, calculate the 11th term using the formula: \[ T_{11} = a + 10d \] Substitute the found values to obtain the 11th term, which is 108. Therefore, option (4) is the correct answer.