Recognize arithmetic and geometric structure, then choose a valid formula for a term or sum.
Why this matters. Sequence questions often hide the model inside a context. Correctly identifying the constant difference or ratio determines every later step.
An arithmetic sequence changes by a constant difference $d$. A geometric sequence changes by a constant ratio $r$. Check consecutive terms before choosing a formula.
$u_n=u_1+(n-1)d$
$S_n=\dfrac n2\left(2u_1+(n-1)d\right)$
$u_n=u_1r^{n-1}$
$S_n=\dfrac{u_1(1-r^n)}{1-r}$, $r\ne1$
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.
Example 2
Geometric growth
A geometric series begins $5+7.5+11.25+\cdots$. Find the sum of its first eight terms.
Here $u_1=5$ and $r=1.5$. Since the number of terms is finite, convergence is irrelevant.
So $S_8\approx246.3$.
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 $S_\infty=8$.
For $u_1+u_1r+u_1r^2+\cdots$, the terms approach zero only when $|r|<1$. Then
If $|r|\ge1$, do not use this formula. The partial sums do not settle to a finite limit.
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. $|-0.8|=0.8<1$. 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.
Five questions to check the main decisions from this lesson.
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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