Independent solution

How to solve this Interest-Principal Decomposition question

Setup

Setup

Use the standard level-payment loan identities for the interest in payment t and the principal in the following payment. Both quantities are expressed per unit payment.

It=1−vn−t+1I_t=1-v^{n-t+1}

Model

Model

Add 1 − v to the power n−t+1 and v to the power n−t. Factoring the common power of v leaves 1 + v to the power n−t times d, where d = 1 − v.

Pt+1=vn−tP_{t+1}=v^{n-t}

Compute

Compute

The requested symbolic sum is therefore 1 + v to the power n−t times d, matching choice D.

X=It+Pt+1=1+vn−t(1−v)=1+vn−tdX=I_t+P_{t+1}=1+v^{n-t}(1-v)=1+v^{n-t}d

Answer

Answer

The calculation gives 1 + v n −t d for interest-principal decomposition, matching published choice D.

X=1+vn−td(D)\boxed{X=1+v^{n-t}d\quad\text{(D)}}