In a geometric progression, the sixth term is 20 and the ninth term is 160. Find the first term. Given that all its terms are positive, find the sum of the first 8 terms of the progression

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The `n`-th term of geometric progression is `a_n=a_(n-1)r` or more generally 

`a_n=a_m r^(n-m)`, `m leq n`.                                                    (1) 

So is you have two elements of geometric progression, this is how you can calculate `r` by using formula (1).

`a_9=a_6r^3`

`160=20r^3`

`r^3=160/20=8`

`r=2`

Now that we have `r` we can use the following formula to calculate the first term `a_1`

`a_n=a_1r^(n-1)`                                                                 (2)

`a_6=a_1r^5`

`20=a_1cdot2^5`

`a_1=20/32=5/8`

Now that we have both `a_1` and `r` we can calculate the sum of the first `n` terms by using the following formula

`S_n=(a_1(r^n-1))/(r-1)`

`S_8=(5/8(2^8-1))/(2-1)=5/8cdot255=1275/8`  <-- Solution

 

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