The coefficient of x^5 in the expansion of (1+x)^(21) +(1+x)^(22) +...+(1+x)^(30) is


Question and Answers Forum

All Questions Topic List
UNKNOWN Questions
Previous in All Question Next in All Question
Previous in UNKNOWN Next in UNKNOWN
Question Number 56095 by gunawan last updated on 10/Mar/19

$The coefficient of x^{5} in the expansion of$ ${(1 + x)}^{21} + {(1 + x)}^{22} + . . . + {(1 + x)}^{30} is$
	Answered by tanmay.chaudhury50@gmail.com last updated on 10/Mar/19

	${(1 + x)}^{n}$ $l e t (r + 1) t h t e r m c o n t a i n s x^{5}$ ${n c}_{r} x^{r} \to s o x^{r} = x^{5} \to r = 5$ $n o w {(1 + x)}^{21} \to 21 c_{5} x^{5}$ ${(1 + x)}^{22} \to 22 c_{5} x^{5}$ $. . .$ $. . .$ $r e q u i r e d a n s w e r i s$ $21 c_{5} + 22 c_{5} + 23 c_{5} + . . . + 30 c_{5}$
	Answered by mr W last updated on 10/Mar/19

	${(1 + x)}^{21} + {(1 + x)}^{22} + . . . + {(1 + x)}^{30}$ $= \frac{{(1 + x)}^{21} [{(1 + x)}^{10} - 1]}{(1 + x) - 1}$ $= \frac{{(1 + x)}^{31} - {(1 + x)}^{21}}{x}$ $c o e f . o f x^{5} t e r m i s :$ $C_{6}^{31} - C_{6}^{21} = 682017$
	Commented by gunawan last updated on 10/Mar/19

	$wow thank you Sir$
	Commented by malwaan last updated on 10/Mar/19

	$w h a t i s t h e f o r m u l a$ $t h a t y o u u s e d ? p l e a s e$
	Commented by mr W last updated on 10/Mar/19

	$S = {(1 + x)}^{21} + {(1 + x)}^{22} + . . . + {(1 + x)}^{30}$ $i s a g e o m e t r i c p r o g r e s s i o n w i t h$ $a_{1} = {(1 + x)}^{21}$ $q = (1 + x)$ $n = 10$ $S = \frac{a_{1} (q^{n} - 1)}{q - 1} = \frac{{(1 + x)}^{21} [{(1 + x)}^{10} - 1]}{(1 + x) - 1}$ $= \frac{{(1 + x)}^{31} - {(1 + x)}^{21}}{x}$ $x^{6} t e r m o f {(1 + x)}^{31} i s C_{6}^{31} x^{6}$ $x^{6} t e r m o f {(1 + x)}^{21} i s C_{6}^{21} x^{6}$ $x^{5} t e r m o f S i s \frac{C_{6}^{31} x^{6} - C_{6}^{21} x^{6}}{x} = (C_{6}^{31} - C_{6}^{21}) x^{5}$
	Commented by malwaan last updated on 11/Mar/19

	$thank you sir$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum