In Simple Interest, the principal remains strictly fixed at $P$ across all periods. In Compound Interest (চক্রবৃদ্ধি সুদ), at the end of each compounding interval (conversion period), the interest accrued is added to the principal. This accumulated sum becomes the new principal for the subsequent period:
| Comparison Parameter | Simple Interest (SI) | Compound Interest (CI) |
|---|---|---|
| Principal | Constant throughout the loan term. | Increases at the end of each conversion period ($P_{k+1} = A_k$). |
| Annual Interest | Remains identical in every single year. | Increases progressively year after year due to interest on interest. |
| First Year Comparison | $ ext{SI}_1 = rac{Pr}{100}$ | $ ext{CI}_1 = rac{Pr}{100}$ (Exactly equal to SI for 1st year). |
| Subsequent Years | Growth is linear: $A = P + rac{Prt}{100}$. | Growth is exponential: $A = P(1 + rac{r}{100})^n$. $ ext{CI} > ext{SI}$. |
Let Principal $= P$, annual rate of interest $= r\%$, and time $= n$ years:
• End of 1st year: Interest $I_1 = rac{P \cdot r \cdot 1}{100}$. Amount $A_1 = P + rac{Pr}{100} = P\left(1 + rac{r}{100}
ight)$.
• For 2nd year: Principal $P_2 = A_1 = P\left(1 + rac{r}{100}
ight)$.
Interest $I_2 = rac{P_2 \cdot r \cdot 1}{100}$. Amount $A_2 = P_2\left(1 + rac{r}{100}
ight) = P\left(1 + rac{r}{100}
ight)^2$.
• Continuing for $n$ years by mathematical induction:
Compound Interest (চক্রবৃদ্ধি সুদ): $ ext{CI} = A - P = P\left[\left(1 + rac{r}{100} ight)^n - 1 ight]$