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: Use the pairing property of an AP.
In an AP, terms equally spaced from the two ends add to the same value; in particular $a_1+a_{10}=a_2+a_9=\dots$, and this common sum equals twice the average term.
Step 2: Use the given sum to find that average.
Since $S_{10}=0$, the average of the 10 terms is $0$, so $a_1+a_{10}=0$.
Step 3: Solve for $a_{10}$.
Since $a_1=a$, we get $a_{10}=-a$, matching option (C).
\[ \boxed{a_{10}=-a} \]