JEE Main 2025 Jan 23 Shift 2, Mathematics Q24: Measure of Dispersion

JEE Main2025Jan 23, Shift 2Mathematics

The variance of the numbers $8, 21, 34, 47, \dots, 320$ , is ________.

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Solution

This is an AP with first term $8$ , common difference $13$ , last term $320$ . Number of terms:

$8 + (n - 1) (13) = 320 \Rightarrow n - 1 = 24 \Rightarrow n = 25.$

Mean $= \frac{8 + 320}{2} = 164$ .

For an AP with first term $a$ , common difference $d$ and $n$ terms, the variance has the closed form $σ^{2} = \frac{( n ^{2} - 1 ) d ^{2}}{12}$ .

$σ^{2} = \frac{( 625 - 1 ) ( 169 )}{12} = \frac{624 \times 169}{12} = 52 \times 169 = 8788.$

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