Sequences • Lesson rewrite

Sequences and Series

Build the main ideas behind arithmetic and geometric sequences, sigma notation, finite sums and infinite geometric series.

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1. Sequence or series?

A sequence is an ordered list of terms. A series is the sum of terms in a sequence.

Sequence\[u_1, u_2, u_3, \ldots\]

Series\[u_1+u_2+u_3+\cdots\]

In IB questions, always decide first whether you are being asked for a term, such as \(u_{20}\), or a sum, such as \(S_{20}\).

An arithmetic sequence has a constant difference. Add or subtract the same amount each time.

\[u_n=u_1+(n-1)d\]

Here \(u_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

For \(5,8,11,14,\ldots\), the first term is \(5\) and the common difference is \(3\). Therefore:

\[u_n=5+(n-1)3=3n+2\]

The 20th term is \(u_{20}=3(20)+2=62\).

A geometric sequence has a constant ratio. Multiply or divide by the same amount each time.

\[u_n=u_1r^{n-1}\]

Here \(r\) is the common ratio. You find it by dividing a term by the previous term.

For \(3,6,12,24,\ldots\), \(u_1=3\) and \(r=2\). Hence:

\[u_n=3\cdot 2^{n-1}\]

Sigma notation is a compact way to write a sum. The expression below means: substitute \(r=1,2,3,4,5\) into \(2r+1\), then add the results.

\[\sum_{r=1}^{5}(2r+1)=3+5+7+9+11=35\]

Sum terms with a constant difference.

Sum terms with a constant ratio.

An infinite geometric series only has a finite sum if the common ratio satisfies \(|r|\lt 1\). This means the terms get smaller and smaller in size.

\[S_{\infty}=\frac{u_1}{1-r},\qquad |r|\lt 1\]

If \(|r|\ge 1\), the series does not converge to a finite value.