Exponential Growth Formula (original) (raw)

Last Updated : 30 Mar, 2026

Exponential growth is a pattern where a quantity increases over time at a constant percentage rate. For example, the number of blogs increased at a monthly rate of about 15% over one year.

The most commonly used version of the exponential formula is:

y = a(1 + r)t

where the beginning value is a, the time is t, the end value is y, and the rate of change is r in decimal form.

**Key Points

• For growth, use 1+r
• For decay, use 1−r
• Convert percentage to decimal (e.g., 5% = 0.05).

**Continuous Exponential Growth

Continuous exponential growth occurs when a quantity increases continuously over time, rather than at fixed intervals like yearly or monthly.

y = aert

where

• a: initial value
• r: growth rate (in decimal)
• t: time
• e: constant ≈ 2.718

**Sample Problems

**Problem 1. A $100 gift card is the first prize in a radio station contest. A name is announced once a day. If the person does not call within 15 minutes, the award will be increased by 2.5 percent the next day. If there are no winners after t days, write an equation to express the value of the gift card in dollars.

The equation for exponential growth is y = a(1 + r)t.

We have, a = 100, r = 2.5% or 0.025

⇒ y = 100(1 + 0.025)t

y = 100(1.025)t

In the equation y = 100(1.025)t, y is the amount of the gift card and t is the number of days since the contest began.

**Problem 2. Suppose that there is no winner after 10 days in the above problem. Determine the value of the gift card.

As per the above problem, y = 100(1.025)t.

Here, t = 10. Then,

y = 100(1.025)10

y = 128.01

The value of gift card in 10 days would be $128.01.

**Problem 3. Since 2000, the cost of attending college has increased by 5% each year. Write an equation for the amount of tuition, t years after 2000 if the tuition in 2000 was $10850.

The equation for exponential growth is y = a(1 + r)t.

We have, a = $10850, r = 5% or 0.05

⇒ y = 10850(1 + 0.05)t

⇒ y = 10850(1.05)t

**Problem 4. What would be the tuition fee in 2015 for the above problem?

As per the above problem, y = 10850(1.025)t.

Here, t = 2015 - 2000 = 15. Then,

y = 10850(1.05)15

⇒ y = $22555

**Problem 5. In 2010, a gym sold 550 memberships. Subscriptions have climbed by 3% per year since then. For t years, write an equation to reflect the number of memberships sold.

The equation for exponential growth is y = a(1 + r)t.

We have, a = 550, r = 3% or 0.03

⇒ y = 550(1 + 0.03)t

⇒ y = 550(1.03)t

In the equation y = 550(1.03)t, y is the number of subscriptions sold and t is the number of years.

**Problem 6. Find the number of memberships sold by the gym in 2020 in the above formula.

As per the above problem, y = 550(1.03)t.

Here, t = 2020 - 2010 = 10. Then,

y = 550(1.03)10

⇒ y = 740 (approx.)

Practice Problems

1. **Population Growth: A city has a population of 50,000 people, and it grows at a rate of 4% per year. What will the population be like after 10 years?
2. **Investment Growth: If you invest $1,000 at an annual interest rate of 5%, how much will the investment be worth after 15 years?
3. **Bacterial Growth: A bacterial culture starts with 200 bacteria and doubles in number every 3 hours. How many bacteria will there be after 12 hours?
4. **Compound Interest: If you deposit $500 into a savings account with an annual compound interest rate of 3%, how much will be in the account after 8 years?
5. **Radioactive Decay: A radioactive substance has a half-life of 10 years. If the initial amount is 100 grams, how much will remain after 30 years?
6. **Debt Repayment: You owe $2,000 on a loan with an annual interest rate of 7%, compounded yearly. How much will you owe after 5 years?
7. **Population Decline: A town's population is decreasing at a rate of 2% per year. If the current population is 80,000, what will it be in 20 years?
8. **Viral Spread: A virus infects 50 people initially and spreads such that the number of infected people triples every 4 days. How many people will be infected after 16 days?
9. **Growth Rate Calculation: If a quantity grows from 1,000 to 2,000 in 5 years, what is the annual growth rate?
10. **Future Value of Investment: An investment of 10,000growsto10,000 grows to 10,000growsto15,000 in 7 years. What is the annual growth rate?