Question

What is the difference between.....

Answer 1

Taking first summation ∑(1+3n) first

First see few terms of this series. To get first term plug n =0 in 1+3n

So for n =0,
`a_(1) = 1 + 3(0) = 1+0 =1`

similarly for next term plug n =1

For n = 1,
`a_(2) = 1 + 3(1) = 1+3 = 4`

For n =2,
`a_(3) = 1 + 3(2) = 1+6 = 7`

So on ,........and last term will be for n =4

For n =4,
`a_(5) = 1 + 3(4) = 1 + 12 = 13`

So we get terms as 1,4,7,.. with constant difference of 3 between succesive terms. So its arithemetic series. Sum formula for arithmetic series is

`S_(n) = (n(a_(1) + a_(n)))/(2)`

where n is total number of terms,
`a_(1)` and
`a_(n)` are first and last term in series

So here n = 5 ( as n is from 0 till 4). so plug value of n as 5,
`a_(1)` as 1 and
`a_(5)` as 13 in sum formula

`S = (5(1 + 13))/(2)`

`S = (5(14))/(2) = (70)/(2) =35`

So sum of first series is 35

Similarly find sum of second series. Plug i values in 3i-5 as shown and find few terms in series

for i =2,
`a_(1) = 3(2) - 5 = 6 - 5 = 1`

for i =3,
`a_(2) = 3(3) - 5 = 9 - 5 = 4`

so on .... and for last term plug i as 6

for i =6,
`a_(5) = 3(6) - 5 = 18 - 5 = 13`

So for sum we will plug n as 5 ( i =2,3,4,5,6 so total 5 terms), then plug
`a_(1)` as -2 and
`a_(5)` as 13 in sum formula

`S = (5(1 + 13))/(2) = (5(14))/(2) = (70)/(2) =35`

So sum of second series is also 35

So difference between sums of both series will be 35-35 =0

Final answer here is 0