Math Help - Arthimetic progression help

1. Arthimetic progression help

7. The 17th term of an AP is 22, and the sum of the first 17 terms is 102. Find
the 1st term and the common difference.

2. Re: Arthimetic progression help

Originally Posted by Benja303
7. The 17th term of an AP is 22, and the sum of the first 17 terms is 102. Find
the 1st term and the common difference.

Every term of the AP is $a_n=a_1+(n-1)d$.

So $a_{17}=a_1+16d=22.$ and $\sum\limits_{k = 1}^{17} {{a_k}} = \sum\limits_{k = 1}^{17} {\left( {{a_1} + (k - 1)d} \right)} = 17{a_1} + \frac{{16 \cdot 17}}{2}d = ?$

3. Re: Arthimetic progression help

The nth term is:

$a_n=a_1+(n-1)d$

The sum of the first n terms is:

$S_n=\frac{n(a_1+a_n)}{2}$

Use the second equation to find $a_1$, then use the first to find $d$. What do you find?

4. Re: Arthimetic progression help

Ohhh thanks mark. I was doing it was N/2 X the rest. I see . The 2 divides the entire thing. cool. Nice whip by the way.

5. Re: Arthimetic progression help

Originally Posted by Benja303
Ohhh thanks mark. I was doing it was N/2 X the rest. I see . The 2 divides the entire thing. cool. Nice whip by the way.
They are actually the same:

$\frac{n(a_1+a_n)}{2}=\frac{n}{2}(a_1+a_n)$

Did you find that $a_1$ is different than the value you cited in your fist post?