JEE Main 2025 Jan 22 Shift 2, Mathematics Q24: Properties Of Binomial Coefficient

JEE Main2025Jan 22, Shift 2Mathematics

If $r = 1 \sum 30 \frac{r ^{2} ( r 30 ) ^{2}}{( r - 1 30 )} = α \times 2^{29}$ , then $α$ is equal to ______.

Your answer

Solution

Use the ratio identity $\frac{( r 30 )}{( r - 1 30 )} = \frac{31 - r}{r}$ .

So $\frac{r ^{2} ( r 30 ) ^{2}}{( r - 1 30 )} = r^{2} \cdot \frac{31 - r}{r} (r 30) = r (31 - r) (r 30)$ .

Split: $r (31 - r) (r 30) = 31 r (r 30) - r^{2} (r 30)$ .

Use $r (r 30) = 30 (r - 1 29)$ , so $\sum_{r = 1}^{30} r (r 30) = 30 \sum_{r = 1}^{30} (r - 1 29) = 30 \cdot 2^{29}$ .

For $\sum r^{2} (r 30)$ : use $r^{2} (r 30) = 30 r (r - 1 29) = 30 [(r - 1) (r - 1 29) + (r - 1 29)] = 30 [29 (r - 2 28) + (r - 1 29)]$ .

Summing: $\sum_{r = 1}^{30} r^{2} (r 30) = 30 [29 \cdot 2^{28} + 2^{29}] = 30 \cdot 2^{28} (29 + 2) = 30 \cdot 31 \cdot 2^{28}$ .

Therefore:

$\sum_{r = 1}^{30} r (31 - r) (r 30) = 31 \cdot 30 \cdot 2^{29} - 30 \cdot 31 \cdot 2^{28}$

$= 30 \cdot 31 \cdot 2^{28} (2 - 1) = 30 \cdot 31 \cdot 2^{28} = 15 \cdot 31 \cdot 2^{29} = 465 \cdot 2^{29}$ .

$∴ α = 465$ .

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