if the binomial co-efficients of three consecutive terms in the expansion of ( a + xⁿ) are in the radio 1:7:42 then find n

In the binomial coefficient of (1+x)ⁿ  the Coefficients of the 5^th, 6^th and 7^th terms are in A.P find all values of n

If p - q is small compared to either p or q, then show that \(n\sqrt { \frac { p }{ q } } =\frac { \left( n+1 \right) p+\left( n-1 \right) q }{ \left( n-1 \right) p+\left( n+1 \right) q } \)
Hence find \(8\sqrt { \frac { 15 }{ 16 } } \)

Find the coefficient of x⁴ in the expansion of \(\frac { 3-4x+{ x }^{ 2 } }{ { e }^{ 2x } } \)

The 2^nd, 3^rd and 4^th terms in the binomial expansion of (x + a)ⁿ are 240, 720 and 1080 for a suitable value of x. Find x, a and n.

Using Binomial theorem, prove that 6ⁿ - 5n always leaves remainder 1 when divided by 25 for all positive integer n.

Find the last two digits of the number 7⁴⁰⁰.

Find the sum of the first n terms of the series \({1\over 1+\sqrt{2}}+{1\over\sqrt{2}+\sqrt{3}}+{1\over\sqrt{3}+\sqrt{4}}+...\)

Find \(\sqrt [ 3 ]{ 65} .\)

Prove that \(\sqrt [ 3 ]{ x^3+7 } -\sqrt [ 3 ]{ x^3+4 } \) is approximately equal to \({1\over x^2}\) when x is large.