The total number of terms in the expression (x + a)^(100) + (x − a)^(100) after simplification is ?


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Question Number 59133 by Tawa1 last updated on 05/May/19

$The total number of terms in the expression {(x + a)}^{100} + {(x - a)}^{100}$ $after simplification is ?$
	Answered by $@ty@m last updated on 05/May/19
	$In (x−a)^n , No. of negative terms= { ((((n+1)/2) if n is odd.)),(((n/2) if n is even.)) :} All negative terms of (x−a)^n cancel with like positive terms of (x+a)^n . ∴ The number of cancelled terms in the expression (x + a)^n + (x − a)^n is { ((2(n+1) −2×((n+1)/2) if n is odd)),((2(n+1)−2×(n/2) if n is even.)) :} = { ((n+1 if n is odd)),((n+2 if n is even.)) :} ∴ The total number of terms in the vexpression (x + a)^n + (x − a)^n after addition of like terms = { ((((n+1)/2) if n is odd)),((((n+2)/2) if n is even.)) :} When n=100 Req.no. of terms=((102)/2)=51.$
	$I n {(x - a)}^{n},$ $N o . o f n e g a t i v e t e r m s = {\begin{cases} \frac{n + 1}{2} i f n i s o d d . \\ \frac{n}{2} i f n i s e v e n . \end{cases}$ $A l l n e g a t i v e t e r m s o f {(x - a)}^{n} c a n c e l$ $w i t h l i k e p o s i t i v e t e r m s o f {(x + a)}^{n} .$ $∴ The number of c a n c e l l e d terms in the expression {(x + a)}^{n} + {(x - a)}^{n} i s$ ${\begin{cases} 2 (n + 1) - 2 \times \frac{n + 1}{2} i f n i s o d d \\ 2 (n + 1) - 2 \times \frac{n}{2} i f n i s e v e n . \end{cases}$ $= {\begin{cases} n + 1 i f n i s o d d \\ n + 2 i f n i s e v e n . \end{cases}$ $∴ The t o t a l number of terms in the v expression {(x + a)}^{n} + {(x - a)}^{n}$ $a f t e r a d d i t i o n o f l i k e t e r m s$ $= {\begin{cases} \frac{n + 1}{2} i f n i s o d d \\ \frac{n + 2}{2} i f n i s e v e n . \end{cases}$ $W h e n n = 100$ $R e q . n o . o f t e r m s = \frac{102}{2} = 51 .$
	Answered by mr W last updated on 05/May/19

	${(x + a)}^{100} = \underset{k = 0}{\sum^{100}} C_{k}^{100} a^{100 - k} x^{k}$ ${(x - a)}^{100} = \underset{k = 0}{\sum^{100}} C_{k}^{100} {(- 1)}^{100 - k} a^{100 - k} x^{k}$ ${(x + a)}^{100} + {(x - a)}^{100} = \underset{k = 0}{\sum^{100}} C_{k}^{100} [1 + {(- 1)}^{100 - k}] a^{100 - k} x^{k}$ $1 + {(- 1)}^{100 - k} = 2 f o r k = 0, 2, . . ., 100 \Rightarrow 51 t e r m s$ $1 + {(- 1)}^{100 - k} = 0 f o r k = 1, 3, . . ., 99 \Rightarrow 50 t e r m s$ $\Rightarrow {(x + a)}^{100} + {(x - a)}^{100} h a s 51 t e r m s .$
	Commented by $@ty@m last updated on 05/May/19

	$T h a n k s f o r c o r r e c t i o n . . . .$
	Commented by Tawa1 last updated on 05/May/19

	$God bless you sir$
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