Find the middle term(s) in the given expansion.
\(\left( \frac { x }{ 3 } +9y \right) ^{ 10 }\)

If P and Q are the sum of odd and even terms in the expansion\(\left( y+b \right) ^{ n }\), then prove that\(\left( y+b \right) ^{ 2n }+\left( y-b \right) ^{ 2n }=2\left( { P }^{ 2 }+Q^{ 2 } \right) \).

Find the coefficient of x⁴ in the expansion of (1+x)ⁿ (1-x)ⁿ

Expand the following expansions.
(9-6x)^-3/2

Using binomial theorem, expand the following expressions.
\((2x+3y)^{ 5 }\)

Find the number of terms in the expansions of following expressions.
(x+3y)²

Evaluate the following terms. 7th term in the expansion of \(\left( 2x+\frac { y }{ 3 } \right) ^{ 15 }.\)

Fine the coefficient of x⁶ in the expansion of (1-2x)^-5/2.

Expand \(\left( 102 \right) ^{ 5 }\)

Find \((x+1)^{ 6 }+(x-1)^{ 6 }.\) Hence, evaluate \((\sqrt { 2+1) } ^{ 6 }+\left( \sqrt { 2-1 } \right) ^{ 6 }.\)

If the integers r (>1), n (>2) and coefficients of (3r)th and (r + 2)nd terms in the expansion of (1 + x)²ⁿ are equal, then prove that n = 2r.

Expand (2+3y)-5 into four terms along with the condition of validity of expansion in each case.
Descending powers of y.

If the binomial expansion of \(\left( c+dy \right) ^{ -2 }\quad is\quad \frac { 1 }{ 4 } -3y+..\) then find the values of c and d.

Find the cube root of 127 upto four decimal places.

Prove that 11ⁿ-10n, when divided by 100, always leave a remainder 1 where n\(\in\)+N

Find the middle terms in the expansions of \(\left( 3-\frac { { x }^{ 3 } }{ 6 } \right) ^{ 7 }\)