6. Let fn (n = 1, 2, 3, ...) be respectively: (a) n²X[²]

Q: Let v = [1] and let z = [¹] x The Projx is equal to k k [21] where

A: Our objective is given below:

Q: Suppose that q 1 is a constant. Calculate the sum n Sn = kqk = q + 2q² + 3q³ +…+nq” k=1

A:

Q: Find a given that ||a i+ (a – 1)j + (a + 1)k|| = 7.

A: Given:

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

A:

Q: Let Sy := {1-(-1):NEN. Find iNf Sy and sup Sy

A:

Q: 6. Find f'(1) if Ax) = nx

A:

Q: [ (ln(7x))²dæ

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: Show that (an) is increasing

A:

Q: If m divides n, show that [GF(pn):GF( pm)] = n/m.

A: Given: m divides nWe need to show that GFpn:GFpm=nm

Q: 22) let XnN20, 036), iF P(X<W) = 0.05 th the value of wis

A: Given that, X~N(μ=20, σ=0.36) To find the value of W when P(X&lt;W)=0.05

Q: If m divides n, show that [GF(pn):GF( pm)] = n/m.

A: Given : m divides n . To prove : GFpn:GFpm=nm.

Q: Indicate the least integer n such that f(x) = (3æ³ + 2x²)(5æ5 + 4æ*) € O(x").

A: Finding f(x) explicitly using multiplication of polynomial. The highest power of x or degree of f…