For the natural numbers m, n, if [(1 - y)^m(1 + y)^n = 1 + a_1y + a_2y^2 + ....

Question

For the natural numbers m, n, if $${(1 - y)^m}{(1 + y)^n} = 1 + {a_1}y + {a_2}{y^2} + .... + {a_{m + n}}{y^{m + n}}$$ and ${a_1} = {a_2} = 10$, then the value of (m + n) is equal to :

Accepted Answer

$${(1 - y)^m}{(1 + y)^n} = 1 + {a_1}y + {a_2}{y^2} + .... + {a_{m + n}}{y^{m + n}}$$

Given, (${a_1} = {a_2} = 10$) ((1 - my + {}^m{C_2}{y^2} + .....)(1 + ny + {}^n{C_2}{y^2} + .....) = 1 + {a_1}y + {a_2}{y^2} + ....\]

$$ \Rightarrow n - m = 10$$ ..... (i)

$$ \Rightarrow {}^m{C_2} + {}^n{C_2} - mn = 10$$...... (ii)

$${{m(m - 1)} \over 2} + {{n(n - 1)} \over 2} - mn = 10$$

$$ \Rightarrow {{{m^2} - m} \over 2} + {{(10 + m)(9 + m)} \over 2} - m(10 + m) = 10$$

$$ \Rightarrow {m^2} - m + {m^2} + 19m + 90 - 2({m^2} + 10m) = 20$$

$$ \Rightarrow 18m + 90 - 20m = 20$$

$$ \Rightarrow 2m = 70$$

$$ \Rightarrow m = 35$$ \& $n = 45$

$m + n = 80$

Solutions · 1