{Use of Tech} A savings plan
James 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.

b.Find a recurrence relation that generates the sequence {Bₙ}.

검증된 단계별 안내

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 variables clearly: Let $B_n$ be the balance after the $n^{th}$ deposit. Given $B_0 = 0$, we want to express $B_n$ in terms of $B_{n-1}$.

Express the interest accumulation: Before the $n^{th}$ deposit, the balance $B_{n-1}$ earns 0.75% interest. This means the balance grows by a factor of $1 + 0.0075$ (since 0.75% = 0.0075 in decimal). So, the balance after interest but before deposit is $B_{n-1} \times (1 + 0.0075)$.

Add the monthly deposit: After the interest is added, James deposits \$100. So, the new balance after the $n^{th}$ deposit is $B_n = B_{n-1} \(\times$ (1 + 0.0075) + 100\).

Write the recurrence relation explicitly: \[ B_n = 1.0075 \times B_{n-1} + 100, \quad \text{with} \quad B_0 = 0. \]

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Recurrence Relations

A recurrence relation defines each term of a sequence using previous terms. In this problem, the balance after each payment depends on the previous balance and the new deposit, making it essential to express Bₙ in terms of Bₙ₋₁. Understanding how to set up such relations helps model the growth of the savings over time.

Compound Interest

Compound interest means interest is earned on both the initial principal and the accumulated interest from previous periods. Here, the account earns 0.75% interest monthly, which is added before the deposit. Recognizing how interest compounds monthly is crucial to correctly formulating the recurrence.

Sequence and Series in Financial Contexts

Sequences represent the balance after each payment, and understanding their behavior helps analyze savings growth. Financial sequences often involve regular deposits and interest accumulation, requiring knowledge of how to combine arithmetic (deposits) and geometric (interest) components in the recurrence.