If Tr denotes the rth term in the expansion of (1+x)n in ascending powers of x ( n being a - Maths - Binomial Theorem

Question

If Tr denotes the rth term in the expansion of
(1+x)n in ascending powers of x ( n being a natural
number), prove that
r(r+1)Tr+2=(n-r+1)(n-r)x2Tr
given hint Tr=nCr-1x
r-1 and
Tr+2=nCr+1xr+1

Priyanka Kedia · Accepted Answer

Given, Tr be the rth tern in the expansion of 1+xn in ascending
powers of x So,Tr=Cr-1nxr-1And, Tr+2=Cr+1nxr+1Now, L H S =rr+1Tr+2
=rr+1Cr+1nxr+1 =rr+1n!r+1!n-r-1!xr+1 =rr+1n!r+1rr-1!n-r-1!xr+1
r+1!=r+1rr-1! =n!r-1!n-r-1!xr+1Multiply and divid n-r+1n-r, we
have, = n-r+1n-r×n!r-1! n-r+1n-rn-r-1!xr+1 =
n-r+1n-r×n!r-1! n-r+1!xr+1 n-r+1!=n-r+1n-rn-r-1! =
n-r+1n-r×n!r-1! n-r+1!x2xr-1 =n-r+1n-rx2 n!r-1! n-r+1!xr-1
=n-r+1n-rx2×Cr-1nxr-1 =n-r+1n-r x2Tr =R H S Hence proved

If Tr denotes the rth term in the expansion of (1+x)n in ascending powers of x ( n being a natural number), prove that r(r+1)Tr+2=(n-r+1)(n-r)x2Tr given hint Tr=nCr-1x r-1 and Tr+2=nCr+1xr+1

If T_r denotes the rth term in the expansion of (1+x)ⁿ in ascending powers of x ( n being a natural number), prove that

r(r+1)T_r+2=(n-r+1)(n-r)x²T_r

given hint T_r=ⁿC_r-1x^_r-1 and T_r+2=ⁿC_r+1x^r+1