Solve (r = 0 to n)∑[(-1)rnCr /3(r+1)] - Maths - Binomial Theorem

Question

Solve (r = 0 to
n)​​∑[(-1)r​
nCr /3(r+1)]

Aarushi Mishra · Accepted Answer

S=∑r=0n -1rCrn3r+1=∑r=0n -1rn!r!n-r!×13r+1=∑r=0n -1rn!r+1r!n-r!×13=∑r=0n -1rn!r+1!n-r!×13=∑r=0n -1rn+1!r+1!n-r!×13n+1=13n+1∑r=0n -1rn+1!r+1!n+1-r+1!=13n+1∑r=0n -1r Cr+1n+1=-13n+1∑r=0n -1r+1  Cr+1n+1=-13n+1-C1n+1+C2n+1-C3n+1+C4n+1-
+-1n+1 Cn+1n+1=-13n+1-C0n+1+C0n+1-C1n+1+C2n+1-C3n+1+C4n+1-
+-1n+1 Cn+1n+1=-13n+1-C0n+1+C0n+1 10
 1n-C1n+1 11  1n-1+C2n+1 12
 1n-2-C3n+1 13  1n-3+C4n+1 14  1n-4-
+-1n+1 Cn+1n+1 1n+1
 10=-13n+1-C0n+1+1-1n=-13n+1-1+0=13n+1

Solve (r = 0 to n)​​∑[(-1)r​ nCr /3(r+1)]

Solve _{(r = 0 to n)}∑[(-1)^r ⁿC_r/3(r+1)]