Independent solution

How to solve this Loan Interest Decomposition question

Setup

Setup

Let X be the present value of the increasing portion of the twenty-five loan payments. Separate the loan into a level 2,500 annuity and the arithmetic-gradient component.

X=100a25‾∣X=100a_{\overline{25}|}

Model

Model

Multiply the loan value by i to obtain first-period interest. The level part contributes 2,500(1 − v²⁵), while the gradient identity supplies the remaining term.

L=2500a25‾∣+100v25(Da)25‾∣L=2500a_{\overline{25}|}+100v^{25}(Da)_{\overline{25}|}

Compute

Compute

After substituting X = 100a-angle-25 and simplifying, first-period interest is 2,500 − Xv²⁵.

I1=iL=2500−Xv25I_1=iL=2500-Xv^{25}

Answer

Answer

The calculation gives 2500 − Xv 25 for loan interest decomposition, matching published choice D.

I1=2500−Xv25(D)\boxed{I_1=2500-Xv^{25}\quad\text{(D)}}