Lump sum vs. annuity.
One deposit or many. Both grow with compound interest, but the formulas look different because of how the deposits stack up over time.
Set $200/month aside in a retirement account at 6% APR. After 30 years of contributions, you've personally deposited $72,000 (200 × 12 × 30). But the account isn't sitting at $72,000.
It's sitting at $200,903.
The extra $128,903 is interest — money the formula earned on top of what you put in. Annuity is the math that takes a stream of small contributions and turns them into something three times their total. The earlier you start the stream, the more compounding has to work with.
One deposit vs. a stream of deposits.
Two bars from the same scenario: $200/month for 30 years at 6%. Contributions are short; the final balance is tall. The gap is what compound interest plus time does over decades.
The future value of a lump sum uses the compound interest formula directly: A = P(1 + r/n)nt. The future value of an annuity with periodic payment M is A = M[(1 + r/n)nt − 1] / (r/n).
Vocabulary you'll see in word problems
- Lump sum One deposit, left to grow. Future value comes from the compound interest formula.
- Annuity A series of equal periodic payments — weekly, monthly, or annually — into a savings or investment account.
- Periodic payment (M) The amount of each regular deposit. The ALEKS dictionary uses M; some textbooks use PMT.
- Future value (A) What the lump sum or annuity will be worth at the end of the time horizon.
- Time horizon How long the money has to grow. Lengthening t beats raising r for long-term saving.
$500/month for 20 years at 6%.
"You contribute $500 a month to an investment account paying 6% APR compounded monthly. After 20 years, what's the balance?"
Identify the four pieces.
M = $500 (each monthly deposit). r = 0.06 (the annual rate, as a decimal). n = 12 (compounding frequency, monthly). t = 20 (years).
Compute the periodic rate.
The annuity formula needs r/n (the rate per compounding period), not just r.
Compute the exponent.
The exponent is n × t — total number of compounding periods.
Plug into the annuity formula.
Apply A = M[(1 + r/n)nt − 1] / (r/n) with the values from above. Compute the parenthesis first, then subtract 1, then divide, then multiply by M.
Sanity check — what did the formula do?
You contributed $500 × 12 × 20 = $120,000 over 20 years. The account is at $231,020. The formula nearly doubled your money — the extra $111,020 is interest. If you'd left the same $120,000 sitting in cash, you'd still have $120,000. That's the difference compounding makes.
Three problems. Lump sum, annuity, head-to-head time horizon.
Deposit $10,000 as a lump sum at 5% APR compounded monthly for 10 years. Find A.
Deposit $200/month at 7% APR compounded monthly for 25 years. Find A.
Two paths to retirement, both at 6% APR monthly. Plan A: $250/month for 30 years. Plan B: $300/month for 25 years. Both contribute $90,000 total. How much does Plan A end with?
Three fast questions before you move on.
Q1. In the annuity formula, the variable M represents...
Q2. Two annuity plans contribute the same total amount over different time horizons. The longer one usually...
Q3. Lump sum future value uses which formula?
Why annuity feels like magic.
Run the annuity formula at scale and the numbers border on absurd. $200 a month for 40 years at 7% lands near $525,000 at retirement. You only put in $96,000 of your own money over those decades — the other $429,000 is interest doing the work. That gap is what compound interest plus time looks like.
The single most expensive financial mistake people make is starting late. Same $200/month at 7%, starting at 35 instead of 25, finishes at $244,000 instead of $525,000. Ten years cost roughly half the final balance. And no, you can't make that up with a higher monthly contribution — to match $525,000 in 30 years, you'd need to deposit $430/month, more than double.
Next: Lesson 6 pulls every formula from this topic — simple, compound, continuous, APY, lump sum, annuity — into one synthesis: a real savings plan with three time horizons and three vehicles.
Continue to Lesson 06Different angle? Need another rep? These are optional — tap any that look helpful.