Arithmetic Progression first term from Sum and Common Difference

Question

The sum of the first 50 elements of a set is 6925 with a common difference of 5. What is the first element of the set

I know how I would usually find the first term of an AP, but I cannot work out how to work out what the 50th term is from the sum of the first 50 terms and then use that information to find the first term.

Through brute force problem solving I have established that 16 is the first term, but there must be an easier way. Any pointers would be grand.

See https://en.wikipedia.org/wiki/Arithmetic_progression#Sum — lab bhattacharjee, Oct 15 '15 at 10:31
Thanks for your comment. Surely that relies on you knowing what the terms are to perform that calculation? or am I missing something. — Dhatsah, Oct 15 '15 at 10:45

score 0 · Accepted Answer · answered Oct 15 '15 at 11:10

Say $a$ is first term,
$d=5$ is the common difference and
$S_n$ is the sum to $n$ terms = $\frac{n}{2}[2a + (n-1)d]$

We know, $n$th term of series = $a + (n-1)d$
Now, it is given that $S_{50} = 6925$

Therefore, $$\frac{50}{2}[2a + (50-1)\cdot 5] = 6925$$ or $$2a + 245 = 277$$ or $$a = 16$$

Hence first term is 16.

Hope this is what you wanted to know.

Arithmetic Progression first term from Sum and Common Difference

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