Let n- N If 1-x n-a0-a1x-a2x2- -anxn and an-3-an-2-an-1 are in AP then

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Proof by mathematical induction

Let nϵ N If...

Question

Let $n ϵ N$ If ${(1 + x)}^{n} = a_{0} + a_{1} x + a_{2} x^{2} + \cdot \cdot + a_{n} x^{n}$ and $a_{n - 3}, a_{n - 2}, a_{n - 1}$ are in AP then

a_{1}, a_{2}, a_{3}

are in AP

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a_{1}, a_{2}, a_{3}

are in HP

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n=7

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n=14

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Solution

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n ϵ N

(1 + x)^{n} = a_{0} x + a_{1} x + a_{2} x^{2} + . . . + a_{n} x^{n},

a_{n - 3}, a_{n - 2}, a_{n - 1},

a_{r} > 0, r ϵ N

a_{1}

a_{2}

a_{3}

a_{2 n}

\frac{a_{1} + a_{2 n}}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{a_{2} + a_{2 n - 1}}{\sqrt{a_{2}} + \sqrt{a_{3}}} + \frac{a_{3} + a_{2 n - 2}}{\sqrt{a_{3}} + \sqrt{a_{4}}} + . . . + \frac{a_{n} + a_{n + 1}}{\sqrt{a_{n}} + \sqrt{a_{n + 1}}}

{(1 + x)}^{n} = 1 + a_{1} x + a_{1} x^{2} + . . . . + a_{n} x^{n}

a_{1}

a_{2}

a_{3}

a_{1}, a_{2}, a_{3}, . . . . . a_{n}

\frac{a_{1}}{a_{2} + a_{3} + . . . a_{n}}, \frac{a_{2}}{a_{1} + a_{3} + . . . . + a_{n}}, \frac{a_{n}}{a_{1} + a_{2} + . . . + a_{n - 1}}

a_{1}, a_{2}, a_{3}, . . . . a_{n}

\frac{a_{1}}{a_{2} + a_{3} + . . . . + a_{n}}, \frac{a_{2}}{a_{1} + a_{3} + . . . . + a_{n}}, . . . ., \frac{a_{n}}{a_{1} + a_{2} + . . . . + a_{n - 1}}

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