If the third term in the expansion of (1/x+ xlog₁₀ x)⁵ is 1, then x=

2.C₀+2².C₁/2+2³.C₃/3+……..+2ⁿ⁺¹.C_n/n+1 =

If n is odd then C₀²-C₁²+C₂²-……….+(-1)ⁿ C_n² =

Coefficient of x⁵⁰ in (1+x)¹⁰⁰⁰+2x(1+x)⁹⁹⁹+3x²(1+x)⁹⁹⁸+……..+1001x¹⁰⁰⁰ is

The midpoint of the line segment joining (2,3,-1), (4,5,3) is

The coefficient of x⁶ in (1+x+x²+x³+x⁴+x⁵)⁶ is

The ratio of the coefficient of x¹⁵ to the term independent of x in (x²+2/x)¹⁵ is

The number of non zero terms in the expansion of (a+b)²⁰ – (a-b)²⁰ is

^mC_r+ ^mC_r-1ⁿC₁+^mC_r-2. ⁿC₂+………..+ ⁿC_r =

Sum of the last 20 coefficients in the expansion of (1+x)³⁹, when expanded in ascending powers of x, is