If 1-1-x-1-x2-1-x3-1-xn-k-0n ak xk- then which of the following is true-A- a n -2- n n -1-2B- a 92- a 82-2 n -1 C 10 n -1 C 10- n -1 C 9C- a k - n C kD- - k -0 n a k -2 n -1-1

The correct option is D ∑_k=0^na_k=2^n+1 1 1+(1+x)+(1+x)^2+(1+x)^3+......+(1+x)^n=(1+x)^n+1 1/x=∑_k=0^na_kx^k So, the coefficient of x^k in (1+x)^n+1 1/x=a_k=^ ...

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Byju's Answer

Standard XII

Mathematics

Multinomial Theorem

If 1+1+x+1+x2...

Question

If $1 + (1 + x) + (1 + x)^{2} + (1 + x)^{3} + . . . . . . + (1 + x)^{n} = \sum_{k = 0}^{n} a_{k} x^{k},$ then which of the following is true?

$a_{n - 2} = \frac{n (n - 1)}{2}$

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$a_{9}^{2} - a_{8}^{2} =^{n + 1} C_{10} (^{n + 1} C_{10} -^{n + 1} C_{9})$

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$a_{k} =^{n} C_{k}$

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$\sum_{k = 0}^{n} a_{k} = 2^{n + 1} - 1$

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Solution

The correct option is D
$\sum_{k = 0}^{n} a_{k} = 2^{n + 1} - 1$

$1 + (1 + x) + (1 + x)^{2} + (1 + x)^{3} + . . . . . . + (1 + x)^{n} = \frac{(1 + x)^{n + 1} - 1}{x} = \sum_{k = 0}^{n} a_{k} x^{k}$

So, the coefficient of $x^{k}$ in $\frac{(1 + x)^{n + 1} - 1}{x} = a_{k} =^{n + 1} C_{k + 1}$

$a_{n - 2} =^{n + 1} C_{n - 1} = \frac{n (n + 1)}{2}$

$a_{9}^{2} - a_{8}^{2} =^{n + 2} C_{10} (^{n + 1} C_{10} -^{n + 1} C_{9})$

$\sum_{k = 0}^{n} a_{k} =^{n + 1} C_{1} +^{n + 1} C_{2} + . . . . +^{n + 1} C_{n + 1} = 2^{n + 1} - 1$

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If 1+(1+x)+(1+x)2+(1+x)3+......+(1+x)n=∑nk=0akxk, then which of the following is true?

The correct option is D ∑nk=0ak=2n+1−1 1+(1+x)+(1+x)2+(1+x)3+......+(1+x)n=(1+x)n+1−1x=∑nk=0akxk So, the coefficient of xk in (1+x)n+1−1x=ak=n+1Ck+1 an−2=n+1Cn−1=n(n+1)2 a29−a28=n+2C10(n+1C10−n+1C9) ∑nk=0ak=n+1C1+n+1C2+....+n+1Cn+1=2n+1−1

If $1 + (1 + x) + (1 + x)^{2} + (1 + x)^{3} + . . . . . . + (1 + x)^{n} = \sum_{k = 0}^{n} a_{k} x^{k},$ then which of the following is true?