| 5.1 |
Arithmetic Progressions
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| Arithmetic progression (AP) is a sequence of numbers such that each term after the first is obtained by adding the previous one with a constant called common difference, \(d\). |
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| Term |
\(T_1\) |
\(T_2\) |
\(T_3\) |
\(T_4\) |
\(T_5\) |
| Sequence |
\(3\) |
\(5\) |
\(7\) |
\(9\) |
\(11\) |
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| Common difference of the above sequence: |
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\(\begin{aligned} d&=T_2-T_1=T_3-T_2\\&=T_4-T_3=T_5-T_4\\&=2. \end{aligned}\)
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| The \(n^{\text{th}}\) term, \(T_n\), of an arithmetic progression is |
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\(\boxed{T_n=a+(n-1)d}\)
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| where \(a=\)first term, \(d=\)common difference |
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| \(\blacksquare\) The sum of the first n terms, \(S_n\), of an arithmetic progression is |
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\(\boxed{S_n=\dfrac{n}{2} \begin{bmatrix} 2a+(n-1)d\end{bmatrix}}\)
\(\boxed{S_n=\dfrac{n}{2} \begin{bmatrix} a+l\end{bmatrix}}\)
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| where \(l=\)last term |
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| The \(n^{\text{th}}\) term, \(T_n\), of an arithmetic progression can also be found using \(S_n-S_{n-1}\). |
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