The 4th and 11th terms of an AP are -1 and 20 respectively. Find the sum of the first 100 terms.
$S_{100} = \frac{100}{2} ( 2a + (100-1)d) $
I always have trouble approaching these type of questions. How do I find the first term, a and the common difference,d ?
On
Between the $4^{th}$ and $11^{th}$ term, you add the common difference $7$ times, hence it must be $3$.
So the first term is $-1-3\cdot(4-1)=-10$, and the hundredth is $20+3\cdot(100-11)=287$.
Now by linearity, the average value of the hundred first terms is also the average of the extremes and
$$S=100\frac{287-10}{2}.$$
Since you know the fourth and eleventh terms, we have $a+3d=-1$ and $a+10d=20$. You can use these two equations to find the values of $a$ and $d$.