Prove that: (1+ x)*** (n+ 1)x C. + + 2 3 n+1

Q: Prove that (2k+1)! > (k+1)!(k+1)k

A: We shall prove this by induction on k. For initial step, put k=1. LHS=2×1+1!=3!=6 RHS=1+1!1+11=4 LHS…

Q: In (5 – x) – In (5 + x) = -In 9

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Q: The following is a valid identity, true or false? P(k): 1+2+22 +23 + .. +2k = (2(k+1)/2) +…

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Q: Find the smallest integer x for which (x4 + 4x)/(x + 2) is O(xx). 01 02

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Q: 9(x + 6) - 41 = 75 Prove x - 62/9)

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Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x > – 1.

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Q: If P (n) is the statement "n + n" is even", and if P(r) is true, then prove P (r + 1) is true.

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Q: Evaluate In x x (4 - In² x) - In 14- In² x | +C - 2 In 14-In² x | +C 12 -dx In | x (4- In²x) | +C In…

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Q: (A) = and (B) = . Find 2A - BO* none of the above

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Q: Prove that: n(n+ 1) r = 2 i=1

A: ∑i=1nr=1+2+3+....................+nhere it is a arithmatic progression sum of n terms of arithmatic…

Q: Prove that 1¹x 2² x 3³ x...xn" < n(n+1) 2 (2n+1) 3

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Q: (b) Show that 1 11 2 k(k + 2)(k + 4) 96 k=1

A: Given series is ∑k=1∞1kk+2k+4

Q: (a) Prove that for all integer n 2 1, 3|n(n² + 11).

A: a.          In the given problem we have to prove that for all integer n≥1, 3|n(n2+11). To prove…

Q: 1.5 Inx - In(x2 - 1) + 2ln(x + 3)

A: Logarithm simplification

Q: a) 1 (x-1)(x+2)(x+4)

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Q: Roblem (6) Drcuss the unik Ganv (a) n+) (). an n+2 nal

A: (a) Given that ∑n=1∞1-1n+2xn+1 on (0,1). Take an=1-1n+2 The formula to find the convergence is,…

Q: n Prove that & (31²-2₁) = 2n³ tn ²-n 2=1 2 Goal: 2 (k+1) ³) + (1+₁)² = (K+1) 2

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Q: Prove the property of the Fibonacci numbers, F(n) = 5(n 4) + 3F(n-5), for n ≥6

A: We have to prove the property of the given Fibonacci numbers F(n) = 5F(n-4) + 3F(n-5) ,  for n≥6

Q: X1 2 4 X1 + x2 If 3 [x3 X4. x3+ X4

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Q: Prove that (n+1)C n is equal to n+1

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Q: x² + x²³²+x² + x³²+x² + x ² 7 ²4 3, 567 Find the coefficient of X18 in I

A: Given equation is x2+x3+x4+x5+x6+x74. To find: Coefficient of x18.

Q: 10 Which term of the expansion of (x²+4) ¹º contains x 20 Term number contains x 20

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Q: Define P(n) to be the assertion that: n(n + 1)(2n + 1) 6 j= (a) Verify that P(3) is true. (b)…

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Q: Show that:

A: Given, nr+nr+1=n+1r+1

Q: Evaluate (z – 2)(z² + 9)

A: We have given an integral;

Q: If P(n) is the statement "n' + n is divisible by 3' Prove that P(3) is true but P(4) is not true.

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