Arithmetic sequences and series

An arithmetic sequence changes by the same additive amount at every step. This fixed amount is the common difference $d$, found using later term minus earlier term: $d=u_{n+1}-u_n$.

If $d>0$, the sequence increases. If $d<0$, it decreases. If $d=0$, it is constant. Arithmetic change is additive; multiplying by the same factor instead produces a geometric pattern.

Check every consecutive difference supplied. One matching pair is not sufficient evidence that an entire data set is arithmetic.

A recurrence relation defines each new term using the previous term. For an arithmetic sequence, state the starting value and the repeated change: $u_1=a$, $u_{n+1}=u_n+d$.

Recurrence notation describes a step-by-step process and works naturally in spreadsheets or programs. It is convenient for generating consecutive terms but inefficient for a distant term such as the 500th.

A recurrence rule is incomplete without an initial value. The rule $u_{n+1}=u_n+4$ describes infinitely many possible sequences until $u_1$ is specified.

To reach term $n$ from term 1, there are $n-1$ equal steps. Therefore the explicit formula is $u_n=a+(n-1)d$. This finds any term directly without generating all earlier terms.

Expanding gives $u_n=dn+(a-d)$, which is linear in $n$. The gradient is $d$, but the vertical intercept at $n=0$ is $a-d$, not usually the first term.

In an application, define what term 1 represents. If a data table starts at year 0, the indexing must match the formula rather than being silently shifted.

Plot term number $n$ horizontally and term value $u_n$ vertically. Arithmetic-sequence points lie on a straight line because equal increases in $n$ produce equal changes $d$ in $u_n$.

Only positive integer term numbers normally belong to the sequence. A GDC may draw a continuous line between points, but values at non-integer positions may have no meaning in the model.

From a regression line $u_n\approx mn+c$, the approximate common difference is $d\approx m$. If indexing begins at $n=1$, the estimated first term is $a\approx m+c$.

When a term value is known but its position is not, set $a+(n-1)d$ equal to the given value and solve for $n$.

Then validate the solution. A term number must be a positive integer. If the algebra produces a non-integer, the target value is not an exact term of that arithmetic sequence.

For threshold language such as “first exceeds,” solve an inequality and select the first integer satisfying the condition.

If two non-consecutive terms $u_p$ and $u_q$ are known, there are $q-p$ equal steps between them. Therefore $d=\frac{(u_q-u_p)}{(q-p)}$. Substitute back to find $a$.

Alternatively, write two equations using the nth-term formula and solve simultaneously. This method generalizes well when parameters or contextual information are involved.

Once $a$ and $d$ are found, verify both given terms. One successful substitution is not enough to detect every sign or indexing error.

An arithmetic series adds consecutive terms of an arithmetic sequence. The sum of the first $n$ terms is $S_n$. Unlike an infinite geometric series, a non-constant arithmetic series does not approach a finite total as $n$ increases indefinitely, so AI HL uses finite sums here.

If the first and last terms are known, the average term is $\frac{(a+u_n)}{2}$. Multiplying this average by the number of terms gives $S_n=\frac{n(a+u_n)}{2}$.

Replacing $u_n$ by $a+(n-1)d$ gives the equivalent formula $S_n=\frac{n[2a+(n-1)d]}{2}$. Choose the form that uses the information provided most directly.

Write the same sum forwards and backwards: $S_n=a+(a+d)+\cdots+u_n$ and $S_n=u_n+(u_n-d)+\cdots+a$.

Adding corresponding positions gives $a+u_n$ every time. There are $n$ such pairs, so $2S_n=n(a+u_n)$, and therefore $S_n=\frac{n(a+u_n)}{2}$.

This pairing explanation prevents formula memorization from becoming disconnected from meaning: an arithmetic-series total equals number of terms multiplied by average of first and last terms.

If $S_n$ is known, substitute into a sum formula. Because $n$ appears both outside and inside the bracket, the resulting equation is often quadratic.

Solve algebraically or with a GDC, then apply the domain: the number of terms must be a positive integer. Reject negative, zero or non-integer solutions unless the context uses an inequality and asks for a threshold.

Substitute the accepted integer back into the sum formula to verify the total.

The Greek capital letter sigma, $\Sigma$, means “add.” In $\sum\limits_{r=1}^{n}[a+(r-1)d]$, the index $r$ takes the integer values $1,2,\ldots,n$, generating and adding the first $n$ arithmetic terms.

The lower limit gives the first index, the upper limit gives the last index, and the expression tells what to add. The number of included integers from lower limit p to upper limit q is q minus p plus 1.

The index letter is a dummy variable: $\sum\limits_{r=1}^{n}$ and $\sum\limits_{k=1}^{n}$ have the same meaning if the expression changes consistently.

Under simple interest, the same interest amount $Pr$ is added each period, where $P$ is principal and $r$ is the interest rate per period as a decimal. The balances therefore form an arithmetic sequence.

After $n$ periods, $A_n=P(1+nr)$. If balances are listed at the end of years 1, 2, 3, and so on, the first listed balance is $P(1+r)$ and the common difference is $Pr$.

Do not confuse simple interest with compound interest. Compound interest multiplies by the same factor each period and is modelled geometrically.

Arithmetic models apply when an amount changes by a fixed number each step: seats per row, regular weekly savings, straight-line depreciation, production quotas, fitness targets or a fixed annual increase.

Define the first term and what one step means. Decide whether the question asks for one period, a cumulative total, a threshold or an unknown parameter.

A negative common difference can model depletion or depreciation, but the model must stop before it predicts impossible values such as negative stock or negative length.

Real measurements rarely have perfectly equal differences. If consecutive changes are close and show no strong curvature, an arithmetic model may still be useful as an approximation.

A simple estimated difference over equally spaced data is $d_{\text{approx}}=\frac{(u_q-u_p)}{(q-p)}$. Using widely separated endpoints averages short-term fluctuations, although a linear regression uses all data and is usually stronger when technology is available.

For a fitted model $u_n\approx a+(n-1)d$, calculate residuals $e_n=\text{observed}-\text{predicted}$. Small residuals with no systematic pattern support the model; a curved residual pattern suggests that constant additive change is unsuitable.

Predictions within the observed interval are generally safer than extrapolation far beyond it. Always mention relevant practical limits or changing conditions.

A sequence table can generate many terms quickly. Enter the term number in one column and the explicit formula in another, or enter the first term and fill the next cells using previous term plus the fixed difference.

A graph of term number against term value should show equally spaced points on a straight line. Use a linear regression for approximately arithmetic data, then interpret its gradient and intercept using the chosen indexing.

A GDC summation command can check a finite sum, and an equation solver can find n from a term or total. However, record the defining formula and apply integer and contextual constraints yourself.

Calculator menus vary, so practise sequence, table, list, sum and regression features on the model allowed in your examinations.

First identify the unit of one step and the meaning of term 1. Next decide whether constant additive change is exact or approximate. Then define a, d and n with units.

Choose $u_n$ for one term and $S_n$ for an accumulated total. Use technology where efficient, but retain unrounded values until the final step.

Finally validate the result: term numbers and counts should be integers, units must match, and the model must stay within practical bounds. Conclude in a sentence that answers the application.

Frequent errors include using $nd$ instead of $(n-1)d$, reversing the sign of $d$, confusing $u_n$ with $S_n$, and counting terms incorrectly when sigma limits do not begin at 1.

Context errors include treating simple interest as compound, rounding a non-integer exact term number, accepting a negative quadratic root, omitting units, or extrapolating an approximate model without comment.

For full communication, state the formula, substitute values, use technology transparently, retain precision, apply constraints and finish with a contextual sentence.

Attempt each question before reading the steps. The progression moves from pattern recognition to exact models, sigma notation, technology interpretation and imperfect real data.

Show formulas and working even when using a GDC. Give money to a sensible currency precision and round approximate model outputs appropriately.

Use this checklist to choose efficiently between a term formula, a sum formula, sigma notation and a technology-supported approximate model.