Arithmetic Progression. Find 1st term and common difference

Question

The seventh term of an A.P. is 15 and the sum of the first seven terms is 42. Find the first term and common difference.

How do I find the first term and common difference with only 1 term given?

Answer 1

Note: use that $$a_7=a_1+6d=15$$ and $$7a_1+21d=42$$

Answer 2

We have $a_7 = 15$ and $\frac{7(a_1 + a_7)}{2} = 42$.

By plugging in $a_7$ into the second equation, we get $a_1 = -3$. It follows clearly that the difference between two terms will be $3$.

Answer 3

Let $d$ the common difference of consecutive terms in the progression. We are given that $a_7=15$ and that $$a_1+\dots+a_7=7\,\frac{a_1+a_7}2=42,\quad\text{so}\quad a_1+a_7=12.$$ On the other hand, the common difference is $\;d=\dfrac{a_7-a_1}6$.

Answer 4

An arithmetic progression has the form $a_n=a_1+(n-1)d $, where $a_n $ is the general term and $d $ is the common difference...

So, $15=a_7=a_1+(7-1)d=a_1+6d $. The sum of the first $7$ terms is $42=(7a_1+21d) $. So $d=\frac {15-a_1}6=\frac{42-7a_1}{21}\iff a_1= -3$. And, $15=-3+6d$, so $ d=3$.