Question

{Use of Tech} A savings planJames begins a savings plan in which he deposits $$100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits 100.$$Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = $0.a. Write the first five terms of the sequence {Bₙ}.

Accepted Answer

Understand the problem setup: James deposits $$100 at the beginning of each month, and the account earns 0.75% interest monthly. The interest is added first, then the deposit is made each month. Define the sequence {Bₙ} where B₀ = 0 represents the balance before any deposits or interest. For each month n, the balance after the nth deposit is Bₙ. Express the recursive formula for Bₙ: The balance after the nth payment is the previous balance Bₙ₋₁ grown by 0.75% interest, plus the new 100 $$deposit. Mathematically, this is:

$$B_{n} = B_{n-1} 	imes (1 + 0.0075) + 100$$ Calculate the first five terms step-by-step using the recursive formula:
- $$B_0 = 0$
- B_1$ = B_0$ 	imes 1.0075 + 100$$
- $$B_2$$ = $$B_1$$ 	imes 1.0075 + 100$$
- B_3$ = B_2$ 	imes 1.0075 + 100$$
- $$B_4$$ = $$B_3$$ 	imes 1.0075 + 100$$
- B_5$ = B_4$ 	imes 1.0075 + 100$$ Write out each term explicitly by substituting the previous term's value, showing how the balance grows month by month with interest and deposits.

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Key Concepts

Recursive Sequences

Compound Interest

Sequence Initialization and Notation