Question

In an A.P., \(a = -3\) and \(S_{17} = 357\). The value of \(a_{17}\) is

• A. \(47\)
• B. \(39\)
• C. \(45\)
• D. \(42\)

Hint:

Use \(S_n = \frac{n}{2}(a + l)\) instead of \(S_n = \frac{n}{2}[2a + (n-1)d]\) when the question involves the last term directly. It saves a lot of algebra.

Answer 1

The Correct Option is C

To determine the value of \(a_{17}\) in an arithmetic progression (A.P.) where the first term \(a = -3\) and the sum of the first 17 terms \(S_{17} = 357\), we need to use the following formulas from arithmetic progression:

• The sum of the first \(n\) terms of an A.P. is given by the formula: \(S_n = \frac{n}{2} \left(2a + (n - 1) d\right)\)
• The \(n\)-th term of an A.P. is given by: \(a_n = a + (n - 1) d\)

We are given:

• \(a = -3\)
• \(S_{17} = 357\)
• \(n = 17\)

Substitute these values into the sum formula to find the common difference \(d\):

\(S_{17} = \frac{17}{2} \left(2(-3) + 16d\right) = 357\)

Simplifying the equation:

\(\frac{17}{2} \left(-6 + 16d\right) = 357\)\(17(-3 + 8d) = 357\)\(-51 + 136d = 357\)\(136d = 408\)\(d = \frac{408}{136} = 3\)

Now, we have found that the common difference \(d\) is 3. To find the 17th term \(a_{17}\), use the formula for the \(n\)-th term:

\(a_{17} = a + 16d\)\(a_{17} = -3 + 16 \times 3\)\(a_{17} = -3 + 48 = 45\)

Thus, the value of \(a_{17}\) is 45.

Therefore, the correct answer is 45.