Question

Solve this arithmetic Progression​

Answer 1

So, here in part a), we are given n terms (no specific no. of terms given), so no. of terms = n , also, the common difference is defined as the difference between the next term to a specific term and the term, i.e
`{\bf{d=a_(n)-a_(n-1)}}`, where d is the common difference, so here d = 49 - 45 = 4

Also, here the last term is 49 let say it the nth term, as no number of terms are given, so using the Arithmetic Progression formula, we can write as ;

`{:\implies \quad \sf a+(n-1)d=49}`

`{:\implies \quad \sf a+(n-1)4=49}`

`{:\implies \quad \sf a+4n-4=49}`

`{:\implies \quad \sf a=49+4-4n=\boxed{\bf{53-4n}}}`

Now, as we assumed there are n terms, so using the sum formula, we can have ;

`{:\implies \quad \sf S_(n)=(n)/(2)(a+a_(n))}`

`{:\implies \quad \sf S_(n)=(n)/(2)(53-4n+49)}`

`{:\implies \quad \sf S_(n)=(n)/(2)(102-4n)}`

`{:\implies \quad \sf S_(n)=n(56-2n)=\boxed{\bf{-2n^(2)+56n}}}`

Now, for part b) we are given a geometric progression with first term = a = 1, common ratio = r = (3/1) = 3, and no. of terms = n = 6, so now putting all values in the sum formula we will have;

`{:\implies \quad \sf S_(6)=\frac{1\{1-(3)^(6)\}}{1-3}}`

`{:\implies \quad \sf S_(6)=(1-729)/(-2)}`

`{:\implies \quad \sf S_(6)=(-728)/(-2)=\boxed{\bf{364}}}`