Question

Question #5edfb

Answer 1

`10923`
See explanation.

Explanation:

So you have a sequence that goes:
3, 8, 13, 18,...

One possibility that I see right away is that `8-3 = 5`
then `13-8 =5`, then `18-13=5`, so perhaps the sequence is just adding 5 to the previous number, starting from 3.

If we call the first entry, `3=k_1`,
and the second entry of sequence, `8=k_2`,
we can rewrite the first and second entries as follows:
`k_1=3`
`k_2=k_1+5`
the third entry would be written:
`k_3=k_2+5`
fourth entry:
`k_4=k_3+5`
...
sixty-sixth entry:
`k_66=k_65+5`

We want the sum of `k_1` to `k_66`, that is, the sum of all the equations shown above.
We now notice that we can rewrite these equations as such:
`k_1=3`
`k_2=3+5`
`k_3=3+5+5`
`k_4=3+5+5+5`
...
`k_i=3+(i-1)*5`

Thus the sum can be written:
`\Sigma_(i=1)^{66} [3+(i-1)*5]`
which can be simplified to:
`\Sigma_(i=1)^{66} 3 + 5*\Sigma_(i=1)^{66} [(i-1)]`
and even to:
`\Sigma_(i=1)^{66} 3 + 5*[\Sigma_(i=1)^{66} i - \Sigma_(i=1)^{66}1]`
so:
`66*3 + 5*[67*66/2 - 66]`
`=10923`

Answer 2

`S_(66)=10923`

Explanation:

`"the sum to n terms of an arithmetic sequence is"`

`•color(white)(x)S_n=n/2[2a+(n-1)d]`

`"where a is the first term and d the common difference"`

`"here "a=3" and "d=18-13=13-8=8-3=5`

`rArrS_(66)=33[(2xx3)+(65xx5)]`

`color(white)(rArrS_(66))=33(6+325)`

`color(white)(rArrS_(66))=10923`