Question

If sum of first ten terms of an A.P. is zero with a as the first term and d, the common difference, which of the following relation is true ?

• A. 10a + 9d = 0
• B. 2a = 9d
• C. \(a_{10} = -a\)
• D. \(a_{10} = a\)

Hint:

An elegant property of an A.P. is that if the sum of its first \(n\) terms is zero, then the average of the terms is zero.
This means the middle terms cancel out in pairs.
For 10 terms, \(a_1 + a_{10} = 0 \implies a_{10} = -a_1 = -a\).
This allows you to write down the answer in seconds without doing any algebra!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The topic of this question is Arithmetic Progressions (A.P.).
We are given an A.P. whose first term is \(a\), the common difference is \(d\), and the sum of its first ten terms is exactly equal to zero.
We need to find the correct algebraic relationship among the choices.

Step 2: Key Formula or Approach:
- The formula for the sum of the first \(n\) terms of an A.P. is:
\[ S_n = \frac{n}{2}[2a + (n - 1)d] \] - For \(n = 10\), we have:
\[ S_{10} = \frac{10}{2}[2a + 9d] = 0 \implies 5[2a + 9d] = 0 \implies 2a + 9d = 0 \] - The formula for the \(n^{\text{th}}\) term of an A.P. is:
\[ a_n = a + (n - 1)d \] - For \(n = 10\), the tenth term \(a_{10}\) is:
\[ a_{10} = a + 9d \] We will use these equations to find the correct relationship between \(a\) and \(a_{10}\).

Step 3: Detailed Explanation:

• Set up the sum of the first 10 terms equal to zero:
\[ S_{10} = \frac{10}{2}[2a + 9d] = 0 \] \[ 5[2a + 9d] = 0 \] Divide both sides by 5:
\[ 2a + 9d = 0 \]

• Express \(9d\) in terms of \(a\):
\[ 9d = -2a \]

• Write down the expression for the 10\(^{\text{th}}\) term of the A.P. (\(a_{10}\)):
\[ a_{10} = a + 9d \]

• Substitute \(9d = -2a\) into the expression for \(a_{10}\):
\[ a_{10} = a + (-2a) \] \[ a_{10} = a - 2a \] \[ a_{10} = -a \]

Step 4: Final Answer:
The true relation is \(a_{10} = -a\).
Therefore, the correct option is (C).