Can you solve this arithmetic problem

The nth term of an arithmetic series, Tn is given by Tn = a + (n-1)d
where a is the first term and d is the common difference.

I'm going to assume you know what a 'common difference' is, generally what an arithmetic series is
and that you don't need proof of any general formula like the one above.

If the 1st term is 13 then a = 13

If the 15th term is 111 then

Tn = a + (n -1 )d
111 = 13 + (15 -1)d --> Note the substitutions of Tn = 111 , a = 13 and n = 15

Rearranging and solving for d, d = 98/14 = 7
The common difference is 7.

The sum to n terms of an arithmetic series is given by Sn = n/2 * (2a + (n-1)d)

And also in some books by Sn = (n/2)(a + l)
Notice though, that in the first 20 terms, which you are trying to find, the last term is ofcourse the 20th term which is given by l = a + (n-1)d where n happens to be 20.

So a + l = a + a + (n-1)d which is ofcourse 2a + (n-1)d, which saves you the trouble of having to separately find the last term as "l" is now out of the equation and it only has a, n and d in it.

This corresponds to S[20] = 20/2 * (13*2 + (20-1)*7)
S[20] = 10(26 + 133) = 10(159) = 1590.

The sum to 20 terms is 1590.

The n'th term of an arithmetic sequence is: a+(n-1)d

We have:

a = 13, n = 15 and a+(n-1)d = 111 Hence:

13+(15-1)d = 111

=> 14d = 98

=> d = 7

Ans: Common difference d = 7

The formula for the sum Sn of arithmetic series is:

Sn = n/2{2a + (n-1)d] Hence:

S_20 = 20/2 [2*13 + (20 - 1)7]

=> S_20 = 10[26 + 133]

=> S_20 = 1590

Ans: The sum of the first 20 terms = 1590

an arithmetic sequence is a sequence of which the terms increase with a fixed amount. This fixed amount is called the common difference, let's denote it with r for now.
This way, the sequence would be as follows:
13 + 0r
13 + 1r
13 + 3r
...
13 + 14 r
From the fact that the 15th term is 111, we can find the value r:
13 + 14 r = 111
and thus: 14 r = 111 - 13 or : r = 98 / 14 = 7

a1 = 13

a15 = 111

an = a1 + (n- 1)d

substitute what you know...then solve for d:

111 = 13 + (14)d
d = 7

The sum of an arithmetic series is in your textbook...make a contribution...look it up !

one part at a time !

id est