Suppose that a1-12 and an+1 is a third of the value of an for all n=1,2,3,4,5.......

Suppose that a₁-12 and a_n+1 is a third of the value of a_n for all n=1,2,3,4,5.... natural numbers.

a) find the first foudn terms fo the sequence

b) find the formula for all values of the sequence

c) compute the exact value of the infinite sum

Expert Solution

milcah answered 2 years ago

Prove: If a1 = b1 mod n and a2 = b2 mod n then (1) a1...

Prove: If a1 = b1 mod n and a2 = b2 mod n then (1) a1 + a2 = b1 + b2 mod n, (2) a1 − a2 = b1 − b2 mod n, and (3) a1a2 = b1b2 mod n.

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A1 = Asset 1 etc. L1 = Liability 1 etc. A1 with value of $3 million...

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1. compute the least non-negative residue of 4^n (mod 9) for n=1,2,3,4,5.... prove that 6*(4^n)=6 (mod...

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Suppose f is a degree n polynomial where f(a1) = b1, f(a2) = b2, · ·...

Suppose f is a degree n polynomial where f(a1) = b1, f(a2) = b2, · · · f(an+1) = bn+1 where a1, a2, · · · an+1 are n + 1 distinct values. We will define a new polynomial g(x) = b1(x − a2)(x − a3)· · ·(x − an+1) / (a1 − a2)(a1 − a3)· · ·(a1 − an+1) + b2(x − a1)(x − a3)(x − a4)· · ·(x − an+1) / (a2 − a1)(a2 − a3)(a2 − a4)·...

python def create_fourier_dataset(x, max_val=7): """ Return the sum of sin(n*x)/n where n = 1,2,3,4,5... max_val Remember,...

python def create_fourier_dataset(x, max_val=7): """ Return the sum of sin(n*x)/n where n = 1,2,3,4,5... max_val Remember, n is a scalar quantity (float/int). x is a NumPy array and you should return an equal length NumPy array :param x: numpy array :param max_val: The maximum value :return: a tuple with two numpy arrays x and the sum """

Show that (1 + 2 +. . .+n)2 > 12 +. . .+ n2, for n...

Show that (1 + 2 +. . .+n)2 > 12 +. . .+ n2, for n ≥ 2.

Present an O(n) algorithm that sorts n positive integer numbers a1, a2, . . . ,...

Present an O(n) algorithm that sorts n positive integer numbers a1, a2, . . . , an which are known to be bounded by n 2 (so ai ≤ n 2 , for every i = 1, . . . , n. Use the idea of Radix Sort (discussed in class and presented in Section 8.3 in the textbook). Illustrate your algorithm by showing on paper similar to Fig. 8.3, page 198 in the textbook (make sure you indicate clearly...

Prove that 3 divides n^3 −n for all n ≥ 1.

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