Find the number of terms in an arithmetic progression whose first term is $5$ , common difference is $4$ , and whose total sum of all terms is $230$ .

medium 2 marks Class 10 · Mathematics · Arithmetic Progressions

Answer

10

The correct answer is $10$ — Since $n$ must be positive integer, $n = 10$ .

Step-by-step solution

Use the sum formula: $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .
Plug $a = 5$ , $d = 4$ , $S_{n} = 230$ : $230 = \frac{n}{2} [2 \times 5 + (n - 1) \times 4]$ .
Simplify the bracket: $2 \times 5 = 10$ , so $[10 + 4 (n - 1)] = 4 n + 6$ .
Rewrite the equation: $230 = \frac{n}{2} (4 n + 6)$ which gives $n (4 n + 6) = 460$ .
Expand: $4 n^{2} + 6 n - 460 = 0$ and divide by $2$ : $2 n^{2} + 3 n - 230 = 0$ .
Solve the quadratic using the discriminant: $Δ = 3^{2} - 4 \times 2 \times (- 230) = 9 + 1840 = 1849 = 4 3^{2}$ .
Compute the positive root: $n = \frac{- 3 + 43}{2 \times 2} = \frac{40}{4} = 10$ .
Since $n$ must be positive integer, $n = 10$ .

Common mistake: Not simplifying the expression inside the brackets correctly before forming the quadratic equation.

Concept tested: finding number of terms from sum