Question

The second term of an P is 15 and the fifth term is double the first term. Find the sum of the first 20 terms of the series.

• A. 750
• B. 810
• C. 800
• D. 850
• E. None of these

Answer 1

The Correct Option is B

Step 1: Understand the given information.
- The second term of an arithmetic progression (AP) is 15.
- The fifth term is double the first term.
We need to find the sum of the first 20 terms of the AP.

Step 2: Use the formula for the nth term of an AP.
The nth term of an arithmetic progression is given by the formula:
\( T_n = a + (n-1) \times d \),
where:
- \( T_n \) is the nth term,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

From the given information, we can set up the following equations:
- The second term is 15: \( T_2 = a + d = 15 \)
- The fifth term is double the first term: \( T_5 = a + 4d = 2a \)

Step 3: Solve for \( a \) and \( d \).
From the first equation: \( a + d = 15 \), we can express \( d \) as:
\( d = 15 - a \).
Now substitute this into the second equation: \( a + 4d = 2a \)
\( a + 4(15 - a) = 2a \)
\( a + 60 - 4a = 2a \)
\( 60 = 5a \)
\( a = 12 \)
Now substitute \( a = 12 \) into \( d = 15 - a \):
\( d = 15 - 12 = 3 \).

Step 4: Use the formula for the sum of the first \( n \) terms of an AP.
The sum of the first \( n \) terms of an AP is given by the formula:
\( S_n = \frac{n}{2} \times [2a + (n-1) \times d] \).
We are asked to find the sum of the first 20 terms, so \( n = 20 \), \( a = 12 \), and \( d = 3 \).
Substituting the values into the formula:
\( S_{20} = \frac{20}{2} \times [2(12) + (20-1) \times 3] \)
\( S_{20} = 10 \times [24 + 57] \)
\( S_{20} = 10 \times 81 = 810 \)

Step 5: Conclusion.
The sum of the first 20 terms of the series is 810.

Final Answer:
The correct option is (B): 810.