Suppose you deposit $1,000 at the end of each quarter for 5 years at an interest rate of 10% compounded monthly

In order to make this calculation, we shall use the concept of present worth. We will calculate A such that the Present Worth from both these stream of cash flows are equal.

Let us first calculate the present worth of quarterly deposits of $1000 for 5 years, or 20 time periods. Now, the yearly interest rate is 10%, so we can take the quarterly interest rate as (10%/4)= 2.5%.

Present Worth = 1000/1.025 + 1000/1.025² + 1000/1.025³ + ...... + 1000/1.025²⁰ = (1000) × (1.025²⁰ - 1)/(0.025 × 1.025²⁰), which is the formula for addition in Geometric Progression.

Hence, present worth = 1000 × (1.6386 - 1)/(0.025 × 1.6386) = $15589.16

Now, let us calculate the present worth of the yearly deposits for 5 years at an annual interest rate of 10%.

Present worth = A/1.10 + A/1.10² + ... A/1.10⁵ = $A × (1.10⁵ - 1)/(0.10 × 1.10⁵) = $A × (1.6105 - 1)/(1.6105 × 0.10) = 3.7908 × $A.

Hence, equating the two present worths, we have,

3.7908 × A = $15589.16 => A = $15589.16/3.7908 = $4112.37

Hence, the required amount A = $4112.37