Review and choose practice

Check your sequence and series skills

Use the source checklist twice: after studying the lecture and again after attempting the tutorial. Its two columns are blank self-assessments, not a claim that reading a page establishes mastery. For each skill, explain the method and try a fresh example without looking at its solution.

• Distinguish finite and infinite sequences. Generate terms from $u_n=f(n)$ for the allowed positive integer indices, recover $u_1=S_1$ and $u_n=S_n-S_{n-1}$ for $n\ge2$, and describe behaviour as the index grows.
• Recognise an AP by a constant successive difference $u_{n+1}-u_n=d$. Identify its first term and finite term count; use $u_n=a+(n-1)d$. Three consecutive AP terms $u,v,w$ satisfy $2v=u+w$.
• Recognise a GP by a constant successive ratio $u_{n+1}/u_n=r$ where the denominator is nonzero. Identify its first term, ratio and finite term count; use $u_n=ar^{n-1}$. Three consecutive nonzero GP terms satisfy $v^2=uw$ and the same consecutive ratio. Keep the earlier qualifications for zero terms.
• Generate a recurrence with a calculator, describe the resulting behaviour, and express a recurrence as an explicit nth-term formula when possible.
• Distinguish finite and infinite series. Write a series in sigma notation, expand a sigma expression, and count $n-m+1$ terms from index $m$ through $n$ inclusive.
• Apply constant factors, sum/difference rules and prefix subtraction; evaluate finite sums with a calculator. Choose AP, GP or supplied square/cube-sum formulae according to the actual summand.

$$\begin{aligned} \sum_{r=m}^{n}cu_r&=c\sum_{r=m}^{n}u_r,\\ \sum_{r=m}^{n}(u_r\pm v_r)&=\sum_{r=m}^{n}u_r\pm\sum_{r=m}^{n}v_r,\\ \sum_{r=m}^{n}u_r&=\sum_{r=k}^{n}u_r-\sum_{r=k}^{m-1}u_r\quad(k<m). \end{aligned}$$

For AP sums, use $S_n=\frac n2[2a+(n-1)d]=\frac n2(a+u_n)$. For finite GP sums, use $S_n=a(1-r^n)/(1-r)=a(r^n-1)/(r-1)$ when $r\ne1$; when $r=1$, use $S_n=na$. For a nonzero infinite GP, check $|r|<1$ before using $S_\infty=a/(1-r)$.

Recognise these special sigma forms and their index endpoints:

$$\begin{aligned} \sum_{r=m}^{n}c&=(n-m+1)c,\\ \sum_{r=1}^{n}r&=\frac{n(n+1)}2,\\ \sum_{r=1}^{n}a^r&=\frac{a(1-a^n)}{1-a}=\frac{a(a^n-1)}{a-1}\quad(a\ne1). \end{aligned}$$

In the last expression, $a$ is both the first term and common ratio. For $a=1$, the sum is $n$. The finite polynomial also gives zero when $a=0$, even if a definition of GP excludes zero ratios. The Student checklist’s number-of-terms symbol is misprinted on page 58; the Teacher copy correctly uses $n$.

Choose practice by question type

The source’s page-59 routing uses lecture example, discussion-question and review-problem numbers. A dash below means the source leaves that cell blank. These are practice suggestions, not source-completion or mastery indicators.

Source correction: both copies print Examples 19, 20 and 21 under GP finite sums and leave the AP finite-sum lecture cell blank. Their actual problems use AP sums, so they belong in the AP row. Examples 24 and 25 provide GP finite-sum practice. The unchanged printed routing remains in the retained native transcription; the table above makes the correction explicit.

Recover the first term separately

A sequence has partial sums $S_n=3\cdot2^n-2$ for positive integers $n$. Recover its first three terms. A student uses $u_n=S_n-S_{n-1}$ for every $n\ge1$ and concludes that the whole sequence is a GP. Locate the mistake.

After your attempt: compare the reasoning

The first term is $u_1=S_1=4$. For $n\ge2$, subtraction gives $u_n=3\cdot2^{n-1}$, so the first three terms are $4,6,12$. Their successive ratios are $3/2$ and $2$, so the whole sequence is not a GP. By definition the empty partial sum is $S_0=0$; extending the supplied positive-index formula to zero would instead give 1. The formula was not stated at zero. Always check the first term as well as the later pattern.

Choose when a repeated payment is recorded

A model loan starts at $B_0=1000$ dollars. At each month end, 1% interest is added, then a payment of $100$ is made. Write a recurrence for the balance just after payment and calculate $B_1,B_2$. How would the recurrence change if payment came before interest? State when this fixed-payment model needs to stop.

After your attempt: compare the reasoning

With interest first, $B_{n+1}=1.01B_n-100$, giving $B_1=910$ and $B_2=819.10$. With payment first, $C_{n+1}=1.01(C_n-100)$, giving $C_1=909$ and $C_2=817.09$. Define the observation time before generating a calculator table. The model assumes fixed monthly interest, no fees and payments that follow this timing. Stop or adjust the final payment when the outstanding amount is less than the scheduled payment; a negative computed balance would represent overpayment, not a continuing debt.

Explain the arithmetic sum by pairing

Write $S_n=a+(a+d)+\cdots+[a+(n-1)d]$ in forward and reverse order. Add the two lines to derive an AP sum formula. Does an odd number of terms require a different formula? Check your argument with $7+10+13+16+19$.

After your attempt: compare the reasoning

Each of the $n$ columns in the two-line sum equals $2a+(n-1)d$. Thus $2S_n=n[2a+(n-1)d]$, so $S_n=n[2a+(n-1)d]/2$. Both copies contain every term even when $n$ is odd; no special formula is needed. For $a=7,d=3,n=5$, the result is $5(14+12)/2=65$. This derives the formula from the terms being added.