<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start"></th><th style="text-align: center; --cell-justify: center">Arithmetic Progression</th><th style="text-align: center; --cell-justify: center">Geometric progression</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Notation</td><td style="text-align: center; --cell-justify: center">First Term: \(a\)Common Difference: \(d\)</td><td style="text-align: center; --cell-justify: center">First Term: \(a\)Common Ratio: \(r\)</td></tr><tr><td rowspan="2" style="text-align: center; --cell-justify: center">\(n^{th}\) term</td><td style="text-align: center; --cell-justify: center">\(T_n = a + (n - 1)d\)</td><td style="text-align: center; --cell-justify: center">\(u_n = ar^{n - 1}\)</td></tr><tr><td colspan="2" style="text-align: center; --cell-justify: center">\(n^{th}\) term \(= S_n - S_{n-1}\)</td></tr><tr><td style="text-align: center; --cell-justify: center">Sum, \(S_n\)</td><td style="text-align: center; --cell-justify: center">\(S_n = \frac{n}{2}[2a + (n - 1)d]\)\( \quad\quad\quad\quad\: = \frac{n}{2}[\text{First term} + \text{Last term}]\)</td><td style="text-align: center; --cell-justify: center">\(S_n = \frac{a(r^n - 1)}{r - 1}\) or \(\frac{a(1 - r^n)}{1 - r}\)</td></tr><tr><td style="text-align: center; --cell-justify: center">Sum to infinity, \(S_{\infty}\)</td><td style="text-align: center; --cell-justify: center">N.A</td><td style="text-align: center; --cell-justify: center">Series is convergent for&nbsp;\(|r| &lt; 1\), \(S_{\infty} = \frac{a}{1 - r}\)</td></tr><tr><td style="text-align: center; --cell-justify: center">Proving</td><td style="text-align: center; --cell-justify: center">\(T_n - T_{n-1} = \text{constant}\)(common difference)</td><td style="text-align: center; --cell-justify: center">\(\frac{U_n}{U_{n-1}} = \text{constant}\)(common ratio)</td></tr><tr><td style="text-align: center; --cell-justify: center">Mean</td><td style="text-align: center; --cell-justify: center">If \(x\), \(y\) and \(z\) are consecutive terms:\(y - x = z - y\)</td><td style="text-align: center; --cell-justify: center">If \(x\), \(y\) and \(z\) are consecutive terms:\(\frac{y}{x} = \frac{z}{y}\)</td></tr></tbody></table></div></div>

Arithmetic Progression

Geometric progression

First Term: \(a\)Common Difference: \(d\)

First Term: \(a\)Common Ratio: \(r\)

\(S_n = \frac{n}{2}[2a + (n - 1)d]\)\( \quad\quad\quad\quad\: = \frac{n}{2}[\text{First term} + \text{Last term}]\)

\(S_n = \frac{a(r^n - 1)}{r - 1}\) or \(\frac{a(1 - r^n)}{1 - r}\)

Series is convergent for \(|r| < 1\), \(S_{\infty} = \frac{a}{1 - r}\)

\(T_n - T_{n-1} = \text{constant}\)(common difference)

\(\frac{U_n}{U_{n-1}} = \text{constant}\)(common ratio)

If \(x\), \(y\) and \(z\) are consecutive terms:\(y - x = z - y\)

If \(x\), \(y\) and \(z\) are consecutive terms:\(\frac{y}{x} = \frac{z}{y}\)

Arithmetic Progression

Arithmetic Progression

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