If C0, C1, C2, C3, ⋅s Are Binomial Coefficients In The Expansion Of (1 ...

• Tardigrade
• Signup
• Login
• Institution
• Exams
• Blog
• Questions

1. Tardigrade
2. Question
3. Mathematics
4. If C0, C1, C2, C3, ⋅s are binomial coefficients in the expansion of (1 + x)n, then (C0/3)- (C1/4)+ (C2/5)- (C3/6)+ ⋅s is equal to

Q. If C0​, C1​, C2​, C3​, ⋯ are binomial coefficients in the expansion of (1+x)n, then 3C0​​−4C1​​+5C2​​−6C3​​+⋯ is equal to 2444 190 KEAMKEAM 2012Binomial Theorem Report Error A

n+11​−n+22​+n+31​

B

n+11​+n+22​−n+31​

C

n+21​−n+21​+n+31​

D

n+12​−n+21​+n+32​

E

n+21​−n+12​+n+31​

Solution:

We know (1−x)n=C0​−C1​x+C2​x2−…+(−1)n(Cn​⋅xn On multiplying both sides by x2, we get (1−x)nx2=C0​x2−C1​x3+C2​x4+… On integrating both sides by taking limit 0 to 1 ∴∫01​(1−x)nx2dx =∫01​(C0​x2−C1​x3+C2​x4+…)dx ∫01​xn(1−x)2dx =[C0​3x3​−C1​4x4​+C2​5x5​−…]01​ ⇒∫01​xn(1+x2−2x)dx=3C0​​−4C1​​+5C2​​−… ⇒3C0​​−4C1​​+5C2​​−… =∫01​(xn+xn+2−2xn+1)dx =[n+1xn+1​+n+3xn+3​−n+22xn+2​]01​ =[n+11​+n+31​−n+22​]