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For geometric progression f( 1 ) = 2 , f( 4 ) = 54 then find f( 3 ) and S( 3 ). [ Calculator, Method and examples ]

Solution:
Your problem `->` For geometric progression f( 1 ) = 2 , f( 4 ) = 54 then find f( 3 ) and S( 3 ).


We know that, `a_n = a × r^(n-1)`

Here `a_1 = 2`

`=> a × r^(1 - 1) = 2`

`=> a × r^0 = 2`

`=> a = 2 ->(1)`



`a_4 = 54`

`=> a × r^(4 - 1) = 54`

`=> a × r^3 = 54 ->(2)`

Solving `(1)` and `(2)`, we get `a = 2` and `r = 3`


We know that, `a_n = a × r^(n-1)`

`a_3 = 2 × 3^(3 - 1)`

`= 2 × 9`

`= 18`

We know that, `S_n = a * (r^n - 1)/(r - 1)`

`:. S_3 = 2 × (3^3 - 1)/(3 - 1)`

`=> S_3 = 2 × (27 - 1)/2`

`=> S_3 = 2 × 26/2`

`=> S_3 = 26`






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