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7.Binomial Theorem
normal
શ્રેણી $aC_0 + (a + b)C_1 + (a + 2b)C_2 + ..... + (a + nb)C_n$ નો સરવાળો મેળવો
જ્યાં $Cr's$ એ $(1 + x)^n, n \in N$ ના વિસ્તરણમાં સહગુણક દર્શાવે છે
A
$(a + 2nb)2^n$
B
$(2a + nb)2^n$
C
$(a +nb)2^{n - 1}$
D
$(2a + nb)2^{n - 1}$
Solution
Simplifying we get
$a\left[{ }^{n} C_{0}+{ }^{n} C_{1}+{ }^{n} C_{2}+\ldots .^{n} C_{n}\right]+b\left[{ }^{n} C_{1}+2{ }^{n} C_{2}+3{ }^{n} C_{3}+\ldots n^{n} C_{n}\right]$
$=a 2^{n}+b\left(\frac{d(1+x)^{n}}{d x}\right)_{x=1}$
$=a 2^{n}+b\left(n(1+x)^{n-1}\right)_{x=1}$
$=a 2^{n}+b\left(n 2^{n-1}\right)$
$=(2 a+b n) 2^{n-1}$
Standard 11
Mathematics
