Answer No. 3 only. Thank you. A. Accumulation and Amount Functions(1) Consider the amount function A(t) = t² + 2t + 3.a. Find the corresponding accumulation function a(t).b. Verify that a(t) satisfies the three properties of an accumulation function.c. Derive an explicit expression for In in terms of n.(2) Suppose that you invest P 4,000 at time 0 into an investment account with an accumulation func-tion of a(t) = at² + 4ß. At time 4, your investment has accumulated to P5,000. Find the accumu-lated value of your investment at time 10.(3) (A) Find a simplified explicit form (in terms of s and t) for the amount of interest earned betweentimes and time t, which is I = A(t) - A(s), where s < t, ifa. In n for some positive integer n.b. In = 2" for some positive integer n. Hint: The geometric sum of k terms is ofor a constant ratio r # 1.ariTotal interest is the sum of the periodic interests, Is+1, Is+2,..., It.1₁ + +1₂1₁Is+1 Is+2mA(s)SA(s+1) A(s+2)s+1 s +2A(t-1) A(z)t-11-pk+11-r

Answered: Image /qna-images/answer/ebee7dcd-889a-43e2-a16b-94fe9b4cd0a9.jpg

(3) (A) Find a simplified explicit form (in terms of s and t) for the amount of interest earned between time s and time t, which is I = A(t)- A(s), where s < t, if a. In n for some positive integer n. b. In = 2" for some positive integer n. Hint: The geometric sum of k terms is o ar = a¹- Sk for a constant ratio r # 1. Total interest is the sum of the periodic interests, Is+1, Is+2,..., It. 1₁ + +1₂ I₁ 0 A(s) S A(+1) A(+2) s+1 s+2 A(t-1) A(n) t-1 t