Let cr denote the binomial coefficient of xr in the expansion of (1 + x)10. If for a, b ∈ R , c1 + 3.2C2 + 5.3C3 + ….upto 10 terms = α * 2 1 1 2 β − 1 ( C 0 + C 1 2 + C 2 3 + . . . . u p t o 1 0 t e r m s ) then the value of a + b is equal to………..

∑ k = 1 1 0 ( 2 k − 1 ) . k . 1 0 C k = 2 ∑ k 2 1 0 C k − ∑ k . 1 0 C k x = 1 ⇒ ∑ k = 0 1 0 k . 1 0 C k = 1 0 . 2 9 x = 1 ⇒ ∑ k 2 1 0 C k = 9 0 . 2 8 + 1 0 . 2 9 = 2 8 ( 9 0 + 2 0 1 ) = 2 8 . 1 1 0 S = 29 . 110 – 10.29 = 29 . 100 S = 29 . 100 2 1 1 . 2 5 2 1 − 1

Let c_r denote the binomial coefficient of x^r in the expansion of (1 + x)¹⁰. If for a, b $\in R,$ c₁ + 3.2C₂ + 5.3C₃ + ….upto 10 terms = $\frac{α * 2^{11}}{2^{β} - 1} (C_{0} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + . . . . u p t o 10 t e r m s)$ then the value of a + b is equal to………..

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Raj Pandey | Contributor-Level 9

5 months ago

$\sum_{k = 1}^{10} (2 k - 1) . k .^{10} C_{k}$

$= 2 \sum {k^{2}}^{10} C_{k} - \sum k .^{10} C_{k}$

$x = 1 \Rightarrow \sum_{k = 0}^{10} k .^{10} C_{k} = 10 . 2^{9}$

$x = 1 \Rightarrow \sum k^{2}^{10} C_{k} = 90 . 2^{8} + 10 . 2^{9} = 2^{8} (90 + 201) = 2^{8} . 110$

S = 2⁹ . 110 – 10.2⁹ = 2⁹ . 100

S = 2⁹ . 100

$\frac{2^{11} . 25}{2^{1} - 1}$

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