In this lesson:
By the end of this lesson, you should be able to:
What is in ?
$200/month for 24 months at monthly interest. Each deposit earns interest for a different number of months:
24 separate calculations — we need a shortcut.
Compute the ratio of any term to the one before it. What is constant?
A geometric sequence is a list of terms with a constant ratio.
A geometric series is the sum of those terms.
counts terms, not multiplications.
For the series :
Think about both before advancing.
For (, , ):
— the sum and nth-term formulas are not interchangeable.
For the series (100 terms):
Adding by hand requires 99 additions.
The question: Can we find using a single algebraic expression?
This is equation (1). The next step multiplies both sides by to create a shifted copy.
Multiplying by shifts every term one position to the right.
Every interior term appears in both — those will cancel when we subtract.
Subtract equation (2) from equation (1):
From the previous step:
Factor both sides:
Divide both sides by , provided :
For the series with , , :
Apply and verify against the direct sum.
Work it out before the next slide.
Both forms give the same result:
Try plugging into the formula:
This is indeterminate — the formula fails.
Why does this happen algebraically?
When , all terms equal , so:
:
Check before applying the formula.
A student writes: "When : "
What is wrong with this reasoning?
Write , , explicitly before computing each sum.
, , :
, , : formula gives ; use
Close your notes. Starting only from:
Derive the formula in four steps.
Compare your derivation with a neighbor. The key move is the first one — what do you multiply both sides by?
Multiplying by shifts the sum — subtraction cancels all but two terms. The restriction is why the trick works.
Click to begin the narrated lesson
Derive geometric series formula