1. Using induction, prove that 2 (^(^^!) ) ² २ 2 3=1 J 3

Q: n Rị |1-Ri Lecen (A) = II [A exp(-AZ,)]“ [exp(-ar)] z.)][exp(-Ar)]" i=1

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Q: Prove by induction E 3₁ = (3^+¹ -1) 2 i=o

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Q: If k >= 2 and x in R^ k prove that there exists y in R^ k such that y ne0 but xy = 0 Is this also…

A: If x has any components equal to 0, then y can be taken to have the corresponding components. equal…

Q: prove for all x,y in R^n: (1+||x+y||) ≥(1+||x||)/(1+||y||)

A: Sol:- The inequality to be proven is: (1 + ||x + y||) ≥ (1 + ||x||)/(1 + ||y||) Let's start by…

Q: Prove that for a # -1, a>0,n EN IS (1+a)^ n >=1+na.

A: Here let A=(1+a)n, a≠-1, a>0, n∈N and B=1+na

Q: Z=5-2(i^1/2) Find modul [z] Arg [z]

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Q: What is the exponent on the x variable of the 3st/nd/rd/th term in the expansion of 17 (x2 +3y)" ?…

A: using formula:-

Q: e3t -1-3t { 9 O 3/(s^2(s-3)) O 1/(s^2(s+3)) O -3/(s^2(s+3)) 1/(s^2(s-3))

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Q: Show that the identity k n = n 72 n k k-1, olds for all integers 1 <k<n and use it to prove that n…

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Q: to Prove that : Ge = { in (+ + ² ) Z tan z =

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Q: Prove-that '(2) r' (₂+1) ¹ (2+ |) T(₂) nd thus derive Legendre's duplication formular 2' (2) T(2:)…

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Q: If n^2+6n+11 = x^2+y^2 for positive integer n, x and y prove [xy/2] is composite. Note: [t]…

A: Given :              n2+6n+11 = x2+y2              t ≤ t

Q: 1. 15x4- 13x²+ a is o(x*) . Hint: Apply theurem I.

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Show and use d(Xn, FCN))<« d(Xn-1 'x) Vny? To prove X = fx) %3D

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Q: Show that the last digit of the Fermat number Fn =2^(2^n)  is 7 for all integers n ≥ 2. [Hint: By…

A: We have to prove that the last digit of the Fermat number Fn=22n is 7 for all integers n≥2. For…

Q: For all a, b in R, show that -max{a^2, b^2} <ab max{a^2, b^2}.

A: To prove that for all (a,b)in(R), [max(−a2b2,ab)≤ab≤max(a2b2)] We'll consider three cases: 1.…

Q: Show thet the Se nng + ベ+」 (+リ(2ペ+)+ ()) (ox) ナ (A+)(2p+1) (P+1) (28+1)(38+) (uhse a>0, B70) (uhse…

A: A series is said  to converge  if  the sequence of partial sums  converge. There are various tests…

Q: Problem 3. Let a, b and n be positive integers. Prove that (a) gcd(an, b″) = gcd(a, b)” and lcm(a”,…

A: To prove the given statements, we will use the following properties of the greatest common divisor…

Q: To prove that dx In /1+ (In x)² + In x + C x/1+ (In x)2 we use the substitutions

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Q: How did they go from 30(2^k-1) to 15(2^k)? Please explain

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Q: Prove that 3! 2(n+3) = = n r=0 (-1) 1 n "Cr r+3 Cr

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Q: . Let n be a positive integer. Prove that 1 · ( n/ 1)  + 2 · ( n/ 2)  + . . . + n ·  (n/ n) …

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Q: [2 4 21 A = l1 3 1 1 1 B = -2 -1 [2 4 D = 4 C = 1 3 21 -1 %3D 2

A: Given A = 242131B = -1111-2-1C = 242131D = 1-1-1-121 Clearly  C = AD = -B

Q: dx (1)(- 2e * )

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Q: n! Verify that Ex1+…+xr=n,xi20 r^n for any r, n>1 x1!x2!...xr!

A: Consider the given information. Let n is greater than equal to 1 arbitrary. Use the mathematical…

Q: Prove algebraically that  A+A'B'=A+B'

A: We have to prove algebraically that  A + A'B' = A + B' We know that, the distributive property  A +…