The sum of 5 numbers in AP is 30 and the sum of their squares is 220. Which of…

For an AP (arithmetic progression) with an odd number of terms, the sum of all the terms equals the number of terms multiplied by the middle term, because the terms pair up symmetrically on either side of the middle term and each such pair adds up to twice the middle term.

Applying this to the given AP of 5 terms:

1. Let the 5 terms be (a − 2d), (a − d), a, (a + d), (a + 2d), where a is the third (middle) term and d is the common difference.

2. Sum of the 5 terms = 5a = 30, so a = 6 — this already gives the third term directly, independent of d.

3. Sum of squares of the 5 terms = 5a2 + 10d2 = 220. Substituting a = 6: 180 + 10d2 = 220, so d2 = 4, giving d = ±2.

4. With a = 6 and d = 2, the AP is 2, 4, 6, 8, 10 (or its reverse for d = −2).

Cross-check: for 2, 4, 6, 8, 10, the sum is 30 and the sum of squares is 4 + 16 + 36 + 64 + 100 = 220, matching both given conditions.

So the third term of the AP is 6.