An arithmetic progression has $9$ terms. Its first term is $2$ and its last term is $50$ . Find the sum of the three middle terms.

medium 2 marks Class 10 · Mathematics · Arithmetic Progressions

Answer

78

The correct answer is $78$ — Add them: $20 + 26 + 32 = 78$ .

Step-by-step solution

For an AP with $n$ terms, the common difference is $d = \frac{l - a}{n - 1}$ where $l$ is the last term.
Compute $d = \frac{50 - 2}{9 - 1} = \frac{48}{8} = 6$ .
The terms are: $t_{1} = 2$ , $t_{2} = 2 + 6$ , $\dots$ , $t_{9} = 50$ .
The three middle terms are $t_{4}$ , $t_{5}$ , $t_{6}$ because $9$ terms have a central term $t_{5}$ .
Calculate:
$t_{4} = a + 3 d = 2 + 3 \times 6 = 20$ ;
$t_{5} = a + 4 d = 2 + 4 \times 6 = 26$ ;
$t_{6} = a + 5 d = 2 + 5 \times 6 = 32$ .
Add them: $20 + 26 + 32 = 78$ .

Common mistake: Assuming the middle three terms are

t_{5}

t_{6}

t_{7}

instead of

t_{4}

t_{5}

t_{6}

for a 9‑term series.

Concept tested: middle terms of an AP