Annuity If x dollars is deposited every 2 weeks (26 times...

Annuity If $x$ dollars is deposited every 2 weeks $(26$ times per year) into an account paying an annual interest rate $r,$ expressed in decimal form, then the amount $A$ in the account after $t$ years can be approximated by the formula $$ A=x\left(\frac{(1+r / 26)^{26 t}-1}{(r / 26)}\right) $$ If $\$ 50$ is deposited every 2 weeks into an account paying $8 \%$ interest, approximate the amount after 10 years.

Annuity If $x$ dollars is deposited every 2 weeks $(26$ times per year) into an account paying an annual interest rate $r,$ expressed in decimal form, then the amount $A$ in the account after $t$ years can be approximated by the formula
$$ A=x\left(\frac{(1+r / 26)^{26 t}-1}{(r / 26)}\right) $$
If $\$ 50$ is deposited every 2 weeks into an account paying $8 \%$ interest, approximate the amount after 10 years.

Key Concepts

Annuity

An annuity is a series of equal payments made at regular intervals over a period of time. The future value of an annuity can be calculated by summing the compounded contributions, where each payment earns interest over a varying number of periods. This concept is essential for understanding how regular deposits grow when combined with compound interest.

Compound Interest

Compound interest is the interest on a deposit calculated based on both the initial principal and the accumulated interest from previous periods. It results in exponential growth since each period's interest is added to the principal for calculating subsequent interest, demonstrating the power of reinvesting earnings.

Periodic Interest Rate Conversion

Period interest rate conversion involves adjusting an annual interest rate to reflect the rate per compounding period. This is done by dividing the annual rate by the number of compounding periods per year, ensuring accurate calculations of interest accrued in each period.

Geometric Series Summation

The annuity formula uses the summation of a geometric series to account for the accumulated contributions. Each deposit is multiplied by a factor representing the compounded growth up to the final period, and the sum of these factors is computed using the geometric series summation formula.

Compounding Frequency

Compounding frequency refers to the number of times per year interest is calculated and added to the account balance. A higher frequency of compounding leads to a greater amount of accumulated interest, as interest is effectively being calculated on an increasingly larger effective principal.

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