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Property 7

If1+x∧ n=C 0+...

Question

If(1+x)^n =C0+C1x+C2x^2+......Cnx^n, prove that

1) 3C0-8C1+13C2-18C3+.....upto (n+1) terms

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Solution

sigma (r=0 to n) (-1)^r(5r+3)ⁿC_r
=5 sigma(r=0 to n)(-1)^rrⁿC_r+3sigma(r=0ton)(-1)^rⁿC_r
=5sigma(r=0 to n)(-1)^rr.(n/r)^n-1C_r-1+3sigma(r=0ton)(-1)^rⁿC_r
=5nsigma(r=0 to n) (-1)^{r n-1}C_r-1+3sigma(r=0ton)(-1)^rⁿC_r
=5n[ -^n-1C₀+^n-1C₁–^n-1C₂+.........(-1)^{n n-1}C_n-1]+ 3 [ⁿC₀-ⁿC₁+....+(-1)ⁿⁿC_n]
=5n(0) +3(0)
=0+0
=0

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Similar questions

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

are in A.P. then show that

a_{1} C_{0} - a_{2} C_{1} + a_{3} C_{2} - . . . . . . + (- 1)^{n} a_{n + 1} C_{n} = 0

Hence deduce that

3 C_{0} - 8 C_{1} + 13 C_{2} - 18 C_{3} + . . . . = 0

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

3 C_{0} + 3^{2} \frac{C_{1}}{2}

\frac{3^{3} C_{2}}{3} + \frac{3^{4} C_{3}}{4}

\frac{3^{n + 1} C_{n}}{n + 1}

\frac{4^{n + 1} - 1}{n + 1}

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

\frac{n!}{x (x + 1) (x + 2) (x + 3) . . . (x + n)} = \frac{C_{0}}{x} - \frac{C_{1}}{x + 1} - . . . + (- 1)^{n} \frac{C_{n}}{x + n}

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

C_{0} C_{r} + C_{1} C_{r + 1} + . . . . . . + C_{n - r} C_{n} = \frac{(2 n)!}{(n + r)! (n - r)!}

n \geq 2

S = C_{0} - 3 C_{1} + 5 C_{2} - 7 C_{3} + . . . .

(n + 1)

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