6. Prove the following identity: n ()-(7)-(-)) (" + ¹) + (x − ¹). k k-1, k

Q: Which of the following is false? O ("2) = 2(") k ㅇ ") %3 (4") + (F) k n(n+1) (":"): 2 2 O ()² = (?)…

A: We have to check whether given statement is true or false.

Q: Let {(=) = In ="(z +2)*(z² + 9)] f'(x) =

A:

Q: 86. Show that Z, = r(cose + isine) and Z2 = r[cos(-6) + isin(-6)] are conjugates. %3!

A: Given,

Q: 133. -Σ 2" Sn+ -(x + 2)" n=0

A: Consider the series, Rewrite this as Now take the constant outside,

Q: Show that (p-1)(p2) (pr) (-1)'r!(mod p), for r= 1, 2, . . . ,p - 1. .

A:

Q: 5.3.2 (a) Suppose that n > 3. Prove that ("+1)ʻ 5.

A: Given:-             n+1n2 < 2 n+1n2 -22<0n+1n+2n+1n-2<0n+1n+2 =…

Q: -Prove that a) b) n -("71) = (-+ 0 (," }) n r r+ n 2-0 (7) = 0 r=0

A: Also we know that    Binomial theorem

Q: 81. Prove: 2+4 +6+ + 2n = n(n + 1). ...

A:

Q: 5. Prove the following identity for n 2 1: Fzn+1 = F+1+ F;

A: We need to prove that ,  F2n + 1 = F2n+1 + F2n We know that ,   Fm+n = FmFn+1 + Fm-1Fn ,  m ≥ 1 , n…

Q: 17. Let P1.Pn be positive real numbers such that P1+ + Pn=1. (a) Prove the following: @P² ≥ (ii)…

A:

Q: Prove that 3 | 4" – 1 where n E N.

A: To prove 3|4n -1  That is 3 divides 4n -1 for every natural number n.

Q: - Evaluate the followings: (i)-r' (4) (ii)-(2) (ii) (-3) Г (iv) Γ(0.5)

A:

Q: If In x + In(x – 2) = ln(7x), then x -

A: From the given problem:   lnx+lnx-2=ln7x   Now using the logarithmic property:   ln(a·b)=ln(a)+ln(b)

Q: Find SU U. 10 9. *+6 x+10 S

A: We have to find length of secent US.

Q: 2. How do you prove a statement of the form ( 3x, P(x) )? Provide examples.

A: To prove a statement of the form (∃x P(x) ), we need to find an example value x such that P(x) is…

Q: If X~ Uniform(b, 1) and o² = 3μz, what is b? (a) -0.82 (b) 12.18 (c) 20.82 (d) 17.00 (e) None of the…

A: To find the value of b, we first need to find expressions for the mean μX​ and…