Sequences and Series

Relationship Between A.M. and G.M.

Tips :

  • The general form of an arithmetic geometric sequence is a, (a + d)r, (a + 2d) r2, (a + 3d)r3........
  • If A and G are the A.M. and G.M., respectively, between two positive numbers, then numbers are A \pm \sqrt{A^{2} - G^{2}}
  • nth term of Arithmetic – geometric progression is [a + (n −1)d] rn−1

Part1: View the Topic in this video From 39:44 To 52:05

Part2: View the Topic in this video From 00:40 To 18:48

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  • nth term of Arithmetic – geometric progression is [a + (n −1)d] rn−1
  • Sum of ‘n’ terms of A.G.P \tt S_{n}=\frac{a}{1-r}+dr\frac{1-r^{n-1}}{\left(1-r\right)^{2}}-\frac{\left[a+\left(n-1\right)d\right]r^{n}}{1-r}, where r ≠ 1 \tt S_{n}=\frac{n}{2}\left[2a+\left(n-1\right)d\right], where r = 1
  • Sum to infinite terms of A.G.P Let |r| < 1. then rn , rn−1 → 0 as n →∝ and it can also be shown that ‘n’ rn → 0 as n → ∝. So, we obtain that \tt S_{n} \rightarrow \frac{a}{1-r}+\frac{dr}{\left(1-r\right)^{2}} , as n → ∝ . In other words, when |r| < 1 the sum to infinity of an arithmetic – geometric series is \tt S_{\propto}=\frac{a}{1-r}+\frac{dr}{\left(1-r\right)^{2}}