By Induction. n Prove that į (3;²_zi) = 2n³ tn ²-n 2 Z=1 2 Goal: (2 (k+1) ³) + (k+₁)² – (k+1) 2 (1) P (1) is true (11) I P(R) is true, then P (k+1) is also the Then, P(n) is true for all in element 2²

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Q: P(n) assert n(n+ 1)(2n + 1) j=1 (a) Verify that P(3) is true (b) Express P(k). (c) Express P(k + 1).

A: Consider the provided question, (a) We have to verify that P(3) is true. Substitute n = 3 in…

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A: topic - functions

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Q: (c) Prove that m is prime if and only if y(m) = m - 1.

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Q: or, Recurrence Formulae : ¨ (2n + 1) xP₂ = (n + 1) Pn+j+nP,-1 (n+1) Pn+1 = (2n + 1) xP₁ − nP,. -

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Q: 1. Using (2n +1)Pn= Pn+1-P-1: prove that P(x) = (2n-1) Pn-1(x) + (2n-5) Pn-2+(2n-9) Pn-3+.... - and…

A: Given : 2n+1Pn=Pn+1'-Pn-1' To Prove : (a) Pn'x=2n-1Pn-1x+2n-5Pn-2+2n-9Pn-3+ . . . . . .…

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A: We will use given formula and value of nr . nr=n!r!×n-r!

Q: 24). Legendre Polynomial P2 (x) · ......... (A) x? (B) 1/2 (x² + 1) (C) 1/2 (3x² - 1) (D) 1/2 (3x² +…

A: We know the legrendre  polynomial expansion.

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Q: 10. Show that an = : 2n5n is also a solution to the recurrence relation an = 7an-1-10an-2. What…

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Q: (11) 11. Prove by induction that 1 1 1+ + + 22 3² + - 2.

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