Dwindling powers

Algebra Level 4

The expression $(1+\sqrt{x})^n$ is expanded in decreasing powers of $x$ . If the fourth term can be expressed in terms of $x^2$ , find the value of $\frac{p}{q+r}$ .

Details:

$p$ = sum of coefficients of all terms except first and last term

$q$ = coefficient of second term

$r$ = coefficient of fourth term

Hint:

You need to find the value of $n$ first.

The answer is 3.

1 solution

Noel Lo
May 3, 2015

The fourth term can be expressed as $nC3(\sqrt{x})^{n-3}$ .

$(\sqrt{x})^{n-3} = x^2$

$(x^{\frac{1}{2}})^{n-3} = x^2$

$x^\frac{n-3}{2} = x^2$

$\frac{n-3}{2} = 2$

$n-3 = 4$

$n=7$

$\frac{p}{q+r}$

= $\frac{7C1+7C2+7C3+7C4+7C5 + 7C6}{7C1+7C3}$ OR $\frac{2^7-7C0 - 7C7}{7C1+7C3}$

= $\frac{7+21+35+35+21+7}{7+35}$ OR $\frac{128-2}{7+35}$

= $\frac{126}{42} = \boxed{3}$

Dwindling powers

The answer is 3.

1 solution

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