12. Series

Learning objectives

12.1Use the binomial theorem• Use the binomial theorem to expand (a + b)^n for positive integer n.• Includes simplification of coefficients.• The formula is given in the List of formulas.12.2Use the general binomial term• Use the general term in a binomial expansion.• Includes finding a specified term such as the term independent of x.• Knowledge of the greatest term and properties of coefficients is not required.12.3Recognise arithmetic and geometric progressions• Recognise arithmetic and geometric progressions and understand the difference between them.12.4Use finite progression formulas• Use formulas for the nth term and the sum of the first n terms to solve problems involving arithmetic or geometric progressions.• Problems may be in context.• Formulas are given in the List of formulas.12.5Use convergence and sum to infinity• Use the condition for convergence of a geometric progression and the formula for the sum to infinity of a convergent geometric progression.• Explain why a particular geometric progression has or does not have a sum to infinity.• The formula is given in the List of formulas.

In $(a+b)^n$ , each term chooses some factors to contribute $b$ and the rest to contribute $a$ . The coefficient ${}^nC_r$ counts the ways to choose the $r$ factors that contribute $b$ .

(a+b)^n=\sum_{r=0}^{n}{}^nC_r,a^{n-r}b^r

As $r$ increases from $0$ to $n$ , the power of $a$ decreases while the power of $b$ increases, and their exponents always add to $n$ . If $b$ is negative, its sign is raised to the power $r$ , so signs alternate when appropriate.

For $(2-3x)^4$ , the five terms are $2^4$ , ${}^4C_1 2^3(-3x)$ , ${}^4C_2 2^2(-3x)^2$ , ${}^4C_3 2(-3x)^3$ and $(-3x)^4$ . After simplifying, the expansion is $16-96x+216x^2-216x^3+81x^4$ .

Here $n$ is a positive integer, so the expansion terminates. The formula is supplied, but every coefficient and power must still be simplified; brackets around a negative term prevent sign errors.

The general term lets you find one required term without writing the whole expansion. In $(a+b)^n$ , choosing $r$ copies of $b$ gives term number $r+1$ , because the first term corresponds to $r=0$ .

T_{r+1}={}^nC_r,a^{n-r}b^r,\qquad 0\le r\le n

Substitute the full expressions for $a$ and $b$ , combine their powers of the variable, and solve the exponent condition. Use exponent $0$ for a term independent of the variable; use the requested exponent for a specified power. Only after finding an integer $r$ in the allowed range should you evaluate the coefficient.

In $\left(6/x^2+x^4/2 ight)^{12}$ , $T_{r+1}={} ^{12}C_r(6x^{-2})^{12-r}(x^4/2)^r$ . The power of $x$ is $-2(12-r)+4r=-24+6r$ . An independent term requires $-24+6r=0$ , so $r=4$ and the term is ${}^{12}C_4 6^8/2^4=51\,963\,120$ .

Do not confuse $r$ with the term number: $r=4$ identifies the fifth term. Greatest-term questions and general properties of binomial coefficients are outside this syllabus objective.

An arithmetic progression changes by a constant difference; a geometric progression changes by a constant multiplier. Test consecutive terms in the same way throughout the sequence.

Feature	Arithmetic progression (AP)	Geometric progression (GP)
constant relationship	$u_{k+1}-u_k=d$	$u_{k+1}/u_k=r$ when the ratio is defined
nth term	$a+(n-1)d$	$ar^{n-1}$
typical pattern	add or subtract the same amount	multiply by the same factor

The sequence $5,9,13,17,\ldots$ is arithmetic because every difference is $4$ . The sequence $12,6,3,3/2,\ldots$ is geometric because every ratio is $1/2$ . The sequence $2,4,7,11,\ldots$ is neither: its differences and ratios both change.

If three algebraic expressions are claimed to form a GP, use $T_2/T_1=T_3/T_2$ , or equivalently $T_2^2=T_1T_3$ when this avoids invalid division. For an AP, use $T_2-T_1=T_3-T_2$ . Then check any excluded zero denominators or parameter conditions.

A steadily increasing sequence is not automatically arithmetic, and a sequence containing powers is not automatically geometric. The invariant difference or ratio is the deciding evidence.

Finite progression problems reduce to the first term $a$ , the common difference $d$ or ratio $r$ , and the number of terms $n$ . Translate every given term or sum before solving for the unknown parameters.

Quantity	Arithmetic progression	Geometric progression
nth term	$u_n=a+(n-1)d$	$u_n=ar^{n-1}$
first $n$ terms	$S_n=\frac n2[2a+(n-1)d]$	$S_n=\frac{a(1-r^n)}{1-r}$ for $r\ne1$
known last term $l$	$S_n=\frac n2(a+l)$	use $l=ar^{n-1}$ if useful

If an AP has third term $10$ and $S_8=116$ , then $a+2d=10$ and $4(2a+7d)=116$ . Solving gives $a=4$ , $d=3$ . A run of 19 terms starting at the 12th ends at the 30th, so its sum is $S_{30}-S_{11}=1216$ .

If a GP has third term $4.5$ and sixth term $15.1875$ , then $ar^2=4.5$ and $ar^5=15.1875$ . Division gives $r^3=3.375$ , so $r=1.5$ and $a=2$ . A block of 10 terms starting at the 16th is $S_{25}-S_{15}$ , not $S_{10}$ .

The nth term is one term, whereas $S_n$ is a cumulative sum. If a GP has $r=1$ , every term is $a$ and $S_n=an$ . For a least-number-of-terms question, solve the inequality and then take the smallest valid positive integer, checking it in the original condition.

A geometric progression has a finite sum to infinity exactly when $|r|<1$ . Then successive terms shrink towards zero, so the remaining tail becomes arbitrarily small.

|r|<1\quad\Longrightarrow\quad S_\infty=\frac{a}{1-r}

From $S_n=a(1-r^n)/(1-r)$ , the condition $|r|<1$ makes $r^n o0$ , leaving $a/(1-r)$ . If $|r|\ge1$ , the terms do not tend to zero, so the partial sums cannot approach a finite limit. Always determine $r$ before using the formula.

For first terms $(2w-1/4),(2w-1/4)^2,(2w-1/4)^3$ , the ratio is $r=2w-1/4$ . Convergence requires $-1<2w-1/4<1$ , giving $-3/8<w<5/8$ . The endpoints are excluded because $|r|=1$ .

If a GP has first term $4$ and its first three terms sum to $7$ , then $4+4r+4r^2=7$ , so $r=1/2$ or $r=-3/2$ . Only $r=1/2$ converges; its sum to infinity is $4/(1-1/2)=8$ .

Check convergence separately for every possible ratio before evaluating $S_\infty$ . An arithmetic progression does not acquire a sum to infinity from these formulas, and a finite geometric sum may exist even when the infinite sum does not.

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12. Series

Learning objectives

Expand a positive-integer binomial

Use the general term to target one binomial term

Tell arithmetic and geometric progressions apart

Model finite arithmetic and geometric progressions

Decide whether a geometric series converges

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