Recognize arithmetic and geometric structure, then choose a valid formula for a term or sum.
Compare what happens from one term to the next. Arithmetic sequences repeat an addition; geometric sequences repeat a multiplication. Whether the terms increase or decrease does not decide which model fits.
$5\to8\to11\to14$
$5\to10\to20\to40$
Check the logic
With first term 4, what happens when the ratio becomes −0.5?
Each multiplication reverses the sign and halves the magnitude.
Set r to −0.5, then 0.5. Compare signs and sizes; explain what the minus sign changes.
Starting at 4 with ratio 0.5 gives 4, 2, 1, 0.5. The partial sums approach 8. A nonzero geometric series has an infinite sum only when |r| < 1.
Call the repeated difference $d$ and the repeated ratio $r$. Starting at term 1, there are $n-1$ steps to term $n$: add $d$ that many times, or multiply by $r$ that many times.
Term 1 needs no step. Term 2 needs one, and term 3 needs two. Counting the moves between terms, rather than the terms themselves, gives $n-1$.
Checkpoint
Correct. Each term is multiplied by $\frac23$. The differences are not constant.
Not quite. Compare ratios as well as differences. A decreasing sequence need not be arithmetic.
Each term is multiplied by $\frac23$. The differences are not constant.
Example 1
Arithmetic model
The fifth term is $17$ and the twelfth term is $45$. Find $u_{30}$.
It removes $u_1$ and isolates the constant difference directly.
A sequence lists terms; a series adds them. For an arithmetic sum, pair the first and last terms: every pair has the same total. For a geometric sum, multiplying by the ratio lines up terms that can cancel.
$S_n=\dfrac n2(2u_1+(n-1)d)$
$S_n=\dfrac{u_1(1-r^n)}{1-r}$ for $r\ne1$; if $r=1$, $S_n=nu_1$.
Write $S_n=u_1+u_1r+\cdots+u_1r^{n-1}$ and multiply by $r$. Subtracting gives $(1-r)S_n=u_1-u_1r^n$. Division by $1-r$ requires $r\ne1$.
Example 2
Geometric growth
A geometric series begins $5+7.5+11.25+\cdots$. Find the sum of its first eight terms.
Check the logic
Here r = 1.5. Can we still find the sum of eight terms?
The condition |r| < 1 is for an infinite sum. Eight terms can always be added.
Compare the finite-sum calculation with the convergence condition later.
Here $u_1=5$ and $r=1.5$. Since the number of terms is finite, convergence is irrelevant.
So $S_8\approx246.3$.
Imagine adding pieces of lengths $4,2,1,1/2,\ldots$. Each piece fills half the remaining gap to 8. The total keeps changing, but the gap tends to zero.
The pieces have lengths 4, 2, 1, 1/2, … . Their partial sums are 4, 6, 7, 7.5, …, approaching 8. After each piece, half the previous gap remains.
For a geometric series with nonzero first term, powers of $r$ shrink to zero exactly when $|r|<1$. In the finite-sum formula, this makes $r^n\to0$, leaving
If $|r|\ge1$ and $u_1\ne0$, the partial sums do not settle to a finite limit. The zero series is the trivial exception.
Negative ratios alternate signs, but the term magnitudes are $|u_1||r|^{n-1}$. They shrink only when $|r|<1$. Terms tending to zero are necessary for any series to converge, but are not sufficient in general; geometric structure is what makes this condition sufficient here.
Example 3
A bouncing ball
A ball is dropped from $12$ m and rebounds to $70\%$ of its previous height. Find the total vertical distance before it settles.
The initial fall contributes $12$. Every rebound height occurs once upward and once downward.
The factor of $2$ is needed for both directions after the first fall.
Checkpoint
Correct. The ratio is $\frac12$, so $|r|<1$ and $S_\infty=8$.
Not quite. An infinite sum exists only for a geometric series with $|r|<1$.
The ratio is $\frac12$, so $|r|<1$ and $S_\infty=8$.
Checkpoint
Correct. $|-0.8|=0.8<1$. Alternating signs do not prevent convergence when the magnitude shrinks.
Not quite. Test $|r|<1$, not merely $r<1$.
$|-0.8|=0.8<1$. Alternating signs do not prevent convergence when the magnitude shrinks.
Your score is not saved.
Arithmetic sequences have a constant first difference.
Arithmetic sequences have a constant first difference.
$u_5=3(2)^4=48$.
$u_5=3(2)^4=48$.
Its ratio is $-0.8$, whose magnitude is less than $1$.
Its ratio is $-0.8$, whose magnitude is less than $1$.
A necessary condition for convergence is that the terms tend to zero.
A necessary condition for convergence is that the terms tend to zero.
Keeping $85\%$ means multiplying by $0.85$ each year.
Keeping $85\%$ means multiplying by $0.85$ each year.
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