Question

How do you find both the 1st term of an arithmetic sequence and the common difference?

for example, when the seventh Term in the sequence is 5 and the sum of the first 16 terms is 20​

Answer 1

An arithmetic sequence is one in which consecutive terms differ by some fixed number. Recursively, such a sequence is given by

`\begin{cases}a_1=a_1\\a_n=a_(n-1)+d&\text{for }n>1\end{cases}`

where
`d` is the common difference between terms.

Using this definition, you can express any term
`a_n` of the sequence in terms of
`a_1`, since

`a_n=a_(n-1)+d`

`a_n=(a_(n-2)+d)+d=a_(n-2)+2d`

`a_n=(a_(n-3)+d)+2d=a_(n-3)+3d`

and so on down to the explicit rule,

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

The sum of the first
`k` terms,
`S_k`, of such a sequence is

`\displaystyle S_k=\sum_(n=1)^ka_n=a_1+a_2+\cdots+a_(k-1)+a_k`

or, using the explicit rule,

`\displaystyle S_k=\sum_(n=1)^k(a_1+(n-1)d)=\sum_(n=1)^k(a_1-d)+d\sum_(n=1)^kn`

`\implies\displaystyle S_k=k(a_1-d)+\frac{dk(k+1)}2=\frac{k(2a_1+(k-1)d)}2`

Now, given
`a_7=5` and
`S_(16)=20`, we have

`a_7=a_1+6d\implies a_1+6d=5`

`S_(16)=\frac{16(2a_1+15d)}2\implies 16a_1+120d=20`

which you can solve to get

`\boxed{a_1=20,d=-\frac52}`