Financial Applications of Arithmetic and Geometric Sequences and Series

GLOSSARY

  • Appreciation – The rise in the value of an asset, such as currency or real estate.

  • Depreciation – The reduction in the value of an asset over its useful life.

  • Compound Interest – The interest earned on both the original investment (principal) and the interest that has already been added over time. It grows faster than simple interest because it earns interest on interest.

  • Loan – Borrowed money to be repaid over time with interest.

  • Mortgage – A type of loan for buying property, such as a house or land. Paid back in regular installments (often monthly) over many years, with interest.

  • Investment – Putting money into something (like a business, stock, or savings plan) with the goal of earning more in the future.

COMPOUND INTEREST

Concept

  • Financial mathematics follows the same patterns in sequences and series.

  • A one-time investment that grows over time follows a geometric sequence.

  • Regular investments with growth form a geometric series.

  • In financial contexts, these are called compound interest and annuities.

  • This lesson focuses on compound interest.

Formula

A = P(1 + i)ⁿ

Where:

  • A = accumulated value

  • P = principal amount

  • i = interest rate per period

  • n = number of periods

Similarities with Geometric Sequence

Geometric Sequence

Compound Interest

Description

an = a₁(rⁿ⁻¹)

A = P(1 + i)ⁿ

Formula

an

A

Total amount / accumulated value

a₁

P

First term / principal amount

r

1 + i

Common ratio

n − 1

n

Geometric starts at n = 1, Compound starts at n = 0

EXAMPLES

Example 1
Mr. Santos invested ₱15,000 in a time deposit account earning 6% annual interest, compounded yearly, for 4 years. How much will it be worth?

Solution:
P = 15,000; i = 0.06; n = 4

A = 15,000(1 + 0.06)⁴
A = ₱18,937.15

Mr. Santos will have ₱18,937.15 after 4 years.

Example 2
You deposit ₱10,000 once in Pag-IBIG MP2 at 7.1% interest and leave it for 5 years. How much will you have?

Solution:
P = 10,000; i = 0.071; n = 5

A = 10,000(1 + 0.071)⁵
A = ₱14,091.18

You will have ₱14,091.18 after 5 years.