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:

Using the formula \(S_n = \frac{n}{2}(a + a_n)\) is much faster than using \(S_n = \frac{n}{2}[2a + (n-1)d]\) because it avoids having to find the common difference \(d\) first!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The topic of this question is Arithmetic Progressions (A.P.).
An Arithmetic Progression is a sequence of numbers with a constant common difference between consecutive terms.
We are given the first term \(a = -3\) and the sum of the first 17 terms \(S_{17} = 357\).
We need to calculate the value of the 17\(^{\text{th}}\) term, \(a_{17}\).

Step 2: Key Formula or Approach:
The sum of the first \(n\) terms of an A.P. is given by the formula:
\[ S_n = \frac{n}{2}(a + a_n) \] where \(a\) is the first term, and \(a_n\) is the \(n^{\text{th}}\) term.
For \(n = 17\), the formula becomes:
\[ S_{17} = \frac{17}{2}(a + a_{17}) \] We will substitute the given values into this equation and solve directly for \(a_{17}\).

Step 3: Detailed Explanation:

• Identify the given parameters:
\(a = -3\)
\(S_{17} = 357\)
\(n = 17\)

• Substitute these values into the sum formula:
\[ 357 = \frac{17}{2}(-3 + a_{17}) \]

• Isolate the term inside the bracket:
Multiply both sides by 2 and divide by 17:
\[ -3 + a_{17} = 357 \times \frac{2}{17} \]

• Perform the division \(357 \div 17\):
We know that \(17 \times 20 = 340\) and \(17 \times 21 = 357\).
\[ \frac{357}{17} = 21 \] Substitute this back:
\[ -3 + a_{17} = 21 \times 2 \] \[ -3 + a_{17} = 42 \]

• Solve for \(a_{17}\):
\[ a_{17} = 42 + 3 = 45 \]

Step 4: Final Answer:
The value of \(a_{17}\) is 45.
Therefore, the correct option is (C).