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AlgebraQuestion and Answers: Page 355

Question Number 22701 Answers: 2 Comments: 0

Question Number 22618 Answers: 1 Comments: 0

If (1 + x)^n = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + ... + C_n x^n , prove that (2^2 /(1.2))C_0 + (2^3 /(2.3))C_1 + (2^4 /(3.4))C_2 + ... + (2^(n+2) /((n + 1)(n + 2)))C_n = ((3^(n+2) − 2n − 5)/((n + 1)(n + 2)))

$${If}\:\left(\mathrm{1}\:+\:{x}\right)^{{n}} \:=\:{C}_{\mathrm{0}} \:+\:{C}_{\mathrm{1}} {x}\:+\:{C}_{\mathrm{2}} {x}^{\mathrm{2}} \:+\:{C}_{\mathrm{3}} {x}^{\mathrm{3}} \\ $$$$+\:...\:+\:{C}_{{n}} {x}^{{n}} ,\:{prove}\:{that} \\ $$$$\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{1}.\mathrm{2}}{C}_{\mathrm{0}} \:+\:\frac{\mathrm{2}^{\mathrm{3}} }{\mathrm{2}.\mathrm{3}}{C}_{\mathrm{1}} \:+\:\frac{\mathrm{2}^{\mathrm{4}} }{\mathrm{3}.\mathrm{4}}{C}_{\mathrm{2}} \:+\:...\:+ \\ $$$$\frac{\mathrm{2}^{{n}+\mathrm{2}} }{\left({n}\:+\:\mathrm{1}\right)\left({n}\:+\:\mathrm{2}\right)}{C}_{{n}} \:=\:\frac{\mathrm{3}^{{n}+\mathrm{2}} \:−\:\mathrm{2}{n}\:−\:\mathrm{5}}{\left({n}\:+\:\mathrm{1}\right)\left({n}\:+\:\mathrm{2}\right)} \\ $$

Question Number 22640 Answers: 0 Comments: 0

With usual notation, show that (C_0 /x) − (C_1 /(x+1)) + (C_2 /(x+2)) − .... + (−1)^n (C_n /(x+n))= ((n!)/(x(x + 1)(x + 2)....(x + n)))

$${With}\:{usual}\:{notation},\:{show}\:{that} \\ $$$$\frac{{C}_{\mathrm{0}} }{{x}}\:−\:\frac{{C}_{\mathrm{1}} }{{x}+\mathrm{1}}\:+\:\frac{{C}_{\mathrm{2}} }{{x}+\mathrm{2}}\:−\:....\:+\:\left(−\mathrm{1}\right)^{{n}} \frac{{C}_{{n}} }{{x}+{n}}= \\ $$$$\frac{{n}!}{{x}\left({x}\:+\:\mathrm{1}\right)\left({x}\:+\:\mathrm{2}\right)....\left({x}\:+\:{n}\right)} \\ $$

Question Number 22612 Answers: 2 Comments: 0

In the binomial expasion of (a − b)^5 , the sum of 2^(nd) and 3^(rd) term is zero, then (a/b) is

$${In}\:{the}\:{binomial}\:{expasion}\:{of}\:\left({a}\:−\:{b}\right)^{\mathrm{5}} , \\ $$$${the}\:{sum}\:{of}\:\mathrm{2}^{{nd}} \:{and}\:\mathrm{3}^{{rd}} \:{term}\:{is}\:{zero}, \\ $$$${then}\:\frac{{a}}{{b}}\:{is} \\ $$

Question Number 22547 Answers: 1 Comments: 0

If α = (5/(2!3)) + ((5.7)/(3!3^2 )) + ((5.7.9)/(4!3^3 )) ,... then find the value of α^2 + 4α.

$$\mathrm{If}\:\alpha\:=\:\frac{\mathrm{5}}{\mathrm{2}!\mathrm{3}}\:+\:\frac{\mathrm{5}.\mathrm{7}}{\mathrm{3}!\mathrm{3}^{\mathrm{2}} }\:+\:\frac{\mathrm{5}.\mathrm{7}.\mathrm{9}}{\mathrm{4}!\mathrm{3}^{\mathrm{3}} }\:,...\:\mathrm{then}\:\mathrm{find} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\alpha^{\mathrm{2}} \:+\:\mathrm{4}\alpha. \\ $$

Question Number 22517 Answers: 0 Comments: 0

Find the coefficient of x in the expansion of [(√(1 + x^2 )) − x]^(−1) in ascending power of x when ∣x∣ < 1.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\left[\sqrt{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\:−\:{x}\right]^{−\mathrm{1}} \:\mathrm{in}\:\mathrm{ascending}\:\mathrm{power} \\ $$$$\mathrm{of}\:{x}\:\mathrm{when}\:\mid{x}\mid\:<\:\mathrm{1}. \\ $$

Question Number 22503 Answers: 1 Comments: 4

If (a + bx)^(−2) = (1/4) − 3x + ..., then (a, b) =

$$\mathrm{If}\:\left({a}\:+\:{bx}\right)^{−\mathrm{2}} \:=\:\frac{\mathrm{1}}{\mathrm{4}}\:−\:\mathrm{3}{x}\:+\:...,\:\mathrm{then}\:\left({a},\:{b}\right)\:= \\ $$

Question Number 22491 Answers: 1 Comments: 0

The coefficient of x^r in the expansion of (1 − 2x)^(−1/2) is (1) (((2r)!)/((r!)^2 )) (2) (((2r)!)/(2^r (r!)^2 )) (3) (((2r)!)/((r!)^2 2^(2r) )) (4) (((2r)!)/(2^r (r + 1)!(r − 1)!))

$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{r}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{1}\:−\:\mathrm{2}{x}\right)^{−\mathrm{1}/\mathrm{2}} \:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\frac{\left(\mathrm{2}{r}\right)!}{\left({r}!\right)^{\mathrm{2}} } \\ $$$$\left(\mathrm{2}\right)\:\frac{\left(\mathrm{2}{r}\right)!}{\mathrm{2}^{{r}} \:\left({r}!\right)^{\mathrm{2}} } \\ $$$$\left(\mathrm{3}\right)\:\frac{\left(\mathrm{2}{r}\right)!}{\left({r}!\right)^{\mathrm{2}} \:\mathrm{2}^{\mathrm{2}{r}} } \\ $$$$\left(\mathrm{4}\right)\:\frac{\left(\mathrm{2}{r}\right)!}{\mathrm{2}^{{r}} \:\left({r}\:+\:\mathrm{1}\right)!\left({r}\:−\:\mathrm{1}\right)!} \\ $$

Question Number 22474 Answers: 0 Comments: 0

Let R = (5(√5) + 11)^(2n+1) and f = R − [R], then prove that Rf = 4^(2n+1) .

$$\mathrm{Let}\:{R}\:=\:\left(\mathrm{5}\sqrt{\mathrm{5}}\:+\:\mathrm{11}\right)^{\mathrm{2}{n}+\mathrm{1}} \:\mathrm{and}\:{f}\:=\:{R}\:−\:\left[{R}\right], \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:{Rf}\:=\:\mathrm{4}^{\mathrm{2}{n}+\mathrm{1}} . \\ $$

Question Number 22472 Answers: 0 Comments: 0

If x^x ∙y^y ∙z^z = x^y ∙y^z ∙z^x = x^z ∙y^x ∙z^y such that x, y and z are positive integers greater than 1, then which of the following cannot be true for any of the possible value of x, y and z? (1) xyz = 27 (2) xyz = 1728 (3) x + y + z = 32 (4) x + y + z = 12

$$\mathrm{If}\:{x}^{{x}} \centerdot{y}^{{y}} \centerdot{z}^{{z}} \:=\:{x}^{{y}} \centerdot{y}^{{z}} \centerdot{z}^{{x}} \:=\:{x}^{{z}} \centerdot{y}^{{x}} \centerdot{z}^{{y}} \:\mathrm{such} \\ $$$$\mathrm{that}\:{x},\:{y}\:\mathrm{and}\:{z}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{greater}\:\mathrm{than}\:\mathrm{1},\:\mathrm{then}\:\mathrm{which}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{cannot}\:\mathrm{be}\:\mathrm{true}\:\mathrm{for}\:\mathrm{any}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:{x},\:{y}\:\mathrm{and}\:{z}? \\ $$$$\left(\mathrm{1}\right)\:{xyz}\:=\:\mathrm{27} \\ $$$$\left(\mathrm{2}\right)\:{xyz}\:=\:\mathrm{1728} \\ $$$$\left(\mathrm{3}\right)\:{x}\:+\:{y}\:+\:{z}\:=\:\mathrm{32} \\ $$$$\left(\mathrm{4}\right)\:{x}\:+\:{y}\:+\:{z}\:=\:\mathrm{12} \\ $$

Question Number 22468 Answers: 0 Comments: 0

If a_r is the coefficient of x^r in the expansion (1 + x + x^2 )^n , then a_1 − 2a_2 + 3a_3 − ....... 2na_(2n) =

$$\mathrm{If}\:{a}_{{r}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{r}} \:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\left(\mathrm{1}\:+\:{x}\:+\:{x}^{\mathrm{2}} \right)^{{n}} ,\:\mathrm{then} \\ $$$${a}_{\mathrm{1}} \:−\:\mathrm{2}{a}_{\mathrm{2}} \:+\:\mathrm{3}{a}_{\mathrm{3}} \:−\:.......\:\mathrm{2}{na}_{\mathrm{2}{n}} \:= \\ $$

Question Number 22463 Answers: 1 Comments: 0

Solve for real x: (1/([x])) + (1/([2x])) = (x) + (1/3), where [x] is the greatest integer less than or equal to x and (x) = x − [x], [e.g. [3.4] = 3 and (3.4) = 0.4].

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:{x}: \\ $$$$\frac{\mathrm{1}}{\left[{x}\right]}\:+\:\frac{\mathrm{1}}{\left[\mathrm{2}{x}\right]}\:=\:\left({x}\right)\:+\:\frac{\mathrm{1}}{\mathrm{3}}, \\ $$$$\mathrm{where}\:\left[{x}\right]\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{less} \\ $$$$\mathrm{than}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to}\:{x}\:\mathrm{and}\:\left({x}\right)\:=\:{x}\:−\:\left[{x}\right], \\ $$$$\left[\mathrm{e}.\mathrm{g}.\:\left[\mathrm{3}.\mathrm{4}\right]\:=\:\mathrm{3}\:\mathrm{and}\:\left(\mathrm{3}.\mathrm{4}\right)\:=\:\mathrm{0}.\mathrm{4}\right]. \\ $$

Question Number 29184 Answers: 1 Comments: 0

{ (((√(x^2 −4xy))+(√(y^2 +2xy+9))=10)),((x−y=7)) :} How many real roots of the equtions system?

$$\begin{cases}{\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{\mathrm{xy}}}+\sqrt{\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{\mathrm{xy}}+\mathrm{9}}=\mathrm{10}}\\{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{y}}=\mathrm{7}}\end{cases} \\ $$$$\boldsymbol{\mathrm{How}}\:\boldsymbol{\mathrm{many}}\:\:\boldsymbol{\mathrm{real}}\:\boldsymbol{\mathrm{roots}}\:\boldsymbol{\mathrm{of}}\: \\ $$$$\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{equtions}}\:\boldsymbol{\mathrm{system}}? \\ $$

Question Number 22423 Answers: 0 Comments: 0

Prove that no three consecutive binomial coefficient can be in G.P. or H.P.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{no}\:\mathrm{three}\:\mathrm{consecutive} \\ $$$$\mathrm{binomial}\:\mathrm{coefficient}\:\mathrm{can}\:\mathrm{be}\:\mathrm{in}\:\mathrm{G}.\mathrm{P}.\:\mathrm{or}\:\mathrm{H}.\mathrm{P}. \\ $$

Question Number 22394 Answers: 0 Comments: 0

Prove that : ((^n C_0 )/n)−((^n C_1 )/(n+1))+((^n C_2 )/(n+2))−...+(−1)^n .((^n C_n )/(2n))=(1/(n.^(2n) C_n ))

$$\mathrm{Prove}\:\mathrm{that}\::\:\frac{\:^{{n}} {C}_{\mathrm{0}} }{{n}}−\frac{\:^{{n}} {C}_{\mathrm{1}} }{{n}+\mathrm{1}}+\frac{\:^{{n}} {C}_{\mathrm{2}} }{{n}+\mathrm{2}}−...+\left(−\mathrm{1}\right)^{{n}} .\frac{\:^{{n}} {C}_{{n}} }{\mathrm{2}{n}}=\frac{\mathrm{1}}{{n}.^{\mathrm{2}{n}} {C}_{{n}} } \\ $$

Question Number 22392 Answers: 0 Comments: 0

Show that the sum of odd coefficients in the expansion of (1 + 2x − 3x^2 )^(1025) is an even integer.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{odd}\:\mathrm{coefficients} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}\:+\:\mathrm{2}{x}\:−\:\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{1025}} \\ $$$$\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{integer}. \\ $$

Question Number 22389 Answers: 0 Comments: 0

Simplify: ^(n−1) C_2 +2^(n−2) C_2 +3^(n−3) C_2 +...+(n−2)^2 C_2

$${Simplify}: \\ $$$$\:^{{n}−\mathrm{1}} {C}_{\mathrm{2}} +\mathrm{2}\:^{{n}−\mathrm{2}} {C}_{\mathrm{2}} +\mathrm{3}\:^{{n}−\mathrm{3}} {C}_{\mathrm{2}} +...+\left({n}−\mathrm{2}\right)\:^{\mathrm{2}} {C}_{\mathrm{2}} \\ $$

Question Number 22379 Answers: 1 Comments: 0

For each positive integer n, define a_n = 20 + n^2 , and d_n = gcd(a_n , a_(n+1) ). Find the set of all values that are taken by d_n and show by examples that each of these values are attained.

$$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:{n},\:\mathrm{define}\:{a}_{{n}} \:= \\ $$$$\mathrm{20}\:+\:{n}^{\mathrm{2}} ,\:\mathrm{and}\:{d}_{{n}} \:=\:{gcd}\left({a}_{{n}} ,\:{a}_{{n}+\mathrm{1}} \right).\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\:\mathrm{values}\:\mathrm{that}\:\mathrm{are}\:\mathrm{taken}\:\mathrm{by} \\ $$$${d}_{{n}} \:\mathrm{and}\:\mathrm{show}\:\mathrm{by}\:\mathrm{examples}\:\mathrm{that}\:\mathrm{each}\:\mathrm{of} \\ $$$$\mathrm{these}\:\mathrm{values}\:\mathrm{are}\:\mathrm{attained}. \\ $$

Question Number 22316 Answers: 0 Comments: 0

Prove that the coefficient of x^p in the expansion of (a_0 +a_1 x+a_2 x^2 +a_3 x^3 +...+a_k x^k )^n is Σ((n!)/(n_0 !n_1 !n_2 !n_3 !...n_k !))a_0 ^n_0 a_1 ^n_1 a_2 ^n_2 a_3 ^n_3 ...a_k ^n_k where n_0 , n_1 , n_2 , n_3 , ..., n_k are all non- negative integers subject to the conditions n_0 +n_1 +n_2 +n_3 +...+n_k =n and n_1 +2n_2 +3n_3 +...+kn_k =p.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{p}} \:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left({a}_{\mathrm{0}} +{a}_{\mathrm{1}} {x}+{a}_{\mathrm{2}} {x}^{\mathrm{2}} +{a}_{\mathrm{3}} {x}^{\mathrm{3}} +...+{a}_{{k}} {x}^{{k}} \right)^{{n}} \\ $$$$\mathrm{is}\:\Sigma\frac{{n}!}{{n}_{\mathrm{0}} !{n}_{\mathrm{1}} !{n}_{\mathrm{2}} !{n}_{\mathrm{3}} !...{n}_{{k}} !}{a}_{\mathrm{0}} ^{{n}_{\mathrm{0}} } {a}_{\mathrm{1}} ^{{n}_{\mathrm{1}} } {a}_{\mathrm{2}} ^{{n}_{\mathrm{2}} } {a}_{\mathrm{3}} ^{{n}_{\mathrm{3}} } ...{a}_{{k}} ^{{n}_{{k}} } \\ $$$$\mathrm{where}\:{n}_{\mathrm{0}} ,\:{n}_{\mathrm{1}} ,\:{n}_{\mathrm{2}} ,\:{n}_{\mathrm{3}} ,\:...,\:{n}_{{k}} \:\mathrm{are}\:\mathrm{all}\:\mathrm{non}- \\ $$$$\mathrm{negative}\:\mathrm{integers}\:\mathrm{subject}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{conditions}\:{n}_{\mathrm{0}} +{n}_{\mathrm{1}} +{n}_{\mathrm{2}} +{n}_{\mathrm{3}} +...+{n}_{{k}} ={n} \\ $$$$\mathrm{and}\:{n}_{\mathrm{1}} +\mathrm{2}{n}_{\mathrm{2}} +\mathrm{3}{n}_{\mathrm{3}} +...+{kn}_{{k}} ={p}. \\ $$

Question Number 22726 Answers: 1 Comments: 2