Question

Find the sum of the first 20 terms. 1.5,1.45,1.40,1.35...

Answer 1

20.5

Explanation:

Answer 2

Let's try to find a pattern to the sequence. You can see that we start from 1.5 and subtract 0.05 to generate the next element with each step. So, if we start from
`a_0=1.5`, the general formula for the n-th element is
`a_n=1.5-0.05n`

So, the sum of the first 20 terms is

`\displaystyle \sum_(n=0)^(19) (1.5-0.05n)`

We can split the sum:

`\displaystyle \sum_(n=0)^(19) 1.5-\sum_(n=0)^(19) (0.05n) = \sum_(n=0)^(19) 1.5-0.05\sum_(n=0)^(19) n`

The first sum is independent of n, so we're just summing 1.5 for 20 times:

`\displaystyle \sum_(n=0)^(19) 1.5= 1.5\cdot 20 = 30`

The second sum is 0.05 times the sum of the first 19 integers. The sum of the first k integers is given by

`(k(k+1))/(2)`

So, the sum of the first 19 integers is

`(19\cdot 20)/(2)=190`

and 0.05 times this sum is

`190\cdot 0.05 = 9.5`

So, the sum of the first 20 elements of the sequence is given by

`\displaystyle \sum_(n=0)^(19) 1.5-0.05\sum_(n=0)^(19) n = 30 - 9.5 = 20.5`