`text(Standard Appearance of An A.G.P. :)`

`S = a + (a + d)r +(a+ 2d)r^2 +(a + 3d)r^3 + ...........................`
Here each terms the product of corresponding terms in an arithmetic and geometric series.


`n^(th)` `text(term of) ` `A.G.P.`

`T_n= [a + (n - 1) d] r^(n-1)`

Where `a =` first term, `d =` common difference, `r= ` common ratio and `n =` position of the term which we require.


`text(Sum of n terms and infinite terms of an )``A.G.P.` :

Let `S=a+(a+d)r+(a+2d)r^2+(a+3d)r^2+...................+(a+(n-1) d)r^(n-1)`

`S.r=+ar+(a+d)r^2+..................+(a+(n-2d)r^(n-1))+(a+(n-1)d)r^n`

`=>S(1-r)=a+dr+dr^2+......................................+dr^(n-1)-[a+(n-1)d]r^n`

`=a+dr((1-r^(n-1))/(1-r))-[a+(n-1)d]r^n`

`S=a/(1-r)+dr((1-r^(n-1))/(1-r)^2)-([a+(n-1)d]r^n)/(1-r)`

If `0<|r|<1` and `n-> oo` then

`r^n,r^(n-1)->0`

`:. S_(oo)=a/(1-r)+(dr)/((1-r)^2)`

Students are suggested not to learn the formula, process should be keep in mind.