In an arithmetic progression, the sum of the first 8 terms is $320$ and the sum of the first 4 terms is $80$ . Find the common difference.

medium 2 marks Class 10 · Mathematics · Arithmetic Progressions

Answer

10

The correct answer is $10$ — Thus the common difference is $10$ .

Step-by-step solution

The sum of the first $n$ terms of an AP is $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .
For $n = 8$ , $S_{8} = \frac{8}{2} [2 a + 7 d] = 4 [2 a + 7 d] = 8 a + 28 d = 320$ (1).
For $n = 4$ , $S_{4} = \frac{4}{2} [2 a + 3 d] = 2 [2 a + 3 d] = 4 a + 6 d = 80$ (2).
Multiply equation (2) by 2: $8 a + 12 d = 160$ .
Subtract this from equation (1): $(8 a + 28 d) - (8 a + 12 d) = 16 d = 160$ .
Divide by 16: $d = 10$ .
Thus the common difference is $10$ .

Common mistake: Using wrong coefficients in the sum formulas

Concept tested: Determine common difference from sums