Question

Charlie has worked hard all summer to earn money. She has found an investment that will pay her 3% each year. At the end of the first year Charlie will have $5,000 in the
Investment
Write a recursive function that will represent Charlie's investment
O m(t) = m(t-1) +0.03, m(1) = 5,000
O m(t) = m(t - 1) +1.03, m(1) = 5,000
O m(t) = m(t-1). 1.03, m(1) = 5,000
x O m(t) = m(t-1)-0.03, m(1) = 5,000

Answer 1

Charlie's returns on the investment is an illustration of a geometric sequence.

The recursive function for Charlie's investment is:

$(c)\ m(t) = m(t-1) \cdot 1.03,\ \ \ m(1) = 5000$

Given that:

$m_1 = 5000$ --- the first investment

$r =0.03$ ---- rate

Charlie's investment at the end of each year are as follows

In terms of the given parameters, each term will be represented as: $m_n = m_1 \times r^{n-1}$

End of the second year

$m_n = m_1 \times r^{n-1}$

$m_2 = 5000 \times (1.03)^{2 - 1}$

$m_2 = 5000 \times (1.03)$

End of the third year

$m_n = m_1 \times r^{n-1}$

$m_3 = 5000 \times (1.03)^{3-1}$

$m_3 = 5000 \times (1.03)^{2}$

Expand

$m_3 = 5000 \times (1.03) \times (1.03)$

Substitute $m_2 = 5000 \times (1.03)$

$m_3 = m_2 \times 1.03$

Express 2 as 3 - 1

$m_3 = m_{3-1} \times 1.03$

Replace 3 with t, to get the expression for t years

$m_t = m_{t-1} \times 1.03$

Hence, the correct recursive expression is:

$(c)\ m(t) = m(t-1) \cdot 1.03,\ \ \ m(1) = 5000$

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