An arithmetic progression has 15 terms. The sum of terms from the 3rd to the 10th term inclusive is $200$ . The first term is $3$ . What is the common difference?

hard MCQ 3 marks Class 10 · Mathematics · Arithmetic Progressions

Answer

4

The correct answer is $4$ — The common difference is $4$ .

Step-by-step solution

The sum of the first $n$ terms is $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .
Sum of the first $10$ terms: $S_{10} = \frac{10}{2} [2 a + 9 d] = 5 [6 + 9 d] = 30 + 45 d$ .
Sum of the first $2$ terms: $S_{2} = \frac{2}{2} [2 a + d] = 1 [6 + d] = 6 + d$ .
Sum of terms $3$ to $10$ equals $S_{10} - S_{2} = (30 + 45 d) - (6 + d) = 24 + 44 d$ .
Set this equal to $200$ : $24 + 44 d = 200$ .
Solve for $d$ : $44 d = 176$ giving $d = 4$ .
The common difference is $4$ .

Common mistake: Subtracting wrong terms or miscalculating indices

Concept tested: Finding common difference from partial sum