# Given a_3=5 and a_5=1 in an arithmetic sequence, what is a_1 and d?

Mar 22, 2018

$\text{First Term " a_1 = 9, " Common Difference } d = - 2$

#### Explanation:

Given ${a}_{3} = 5 , {a}_{5} = 1 , \text{ To find First Term " a_1, " common difference } d$

${a}_{3} = {a}_{1} + \left(3 - 1\right) \cdot d$

${a}_{1} + 2 d = 5 , \text{ Eqn 1 }$

${a}_{5} = {a}_{1} + \left(5 - 1\right) \cdot d$

${a}_{1} + 4 d = 1 , \text{ Eqn 2 }$

Solving equations (1), (2)

$\text{ Eqn 2 - Eqn 1 }$

$4 d - 2 d = 1 - 5 = - 4$

$2 d = - 4 \text{ or } d = - 2$

Substituting value of d in Eqn (1),

${a}_{1} + 2 \cdot - 2 = 5$

${a}_{1} = 5 + 4 = 9$

Mar 22, 2018

${a}_{1} = 9$ and $d = - 2$.

#### Explanation:

${U}_{n} = a + \left(n - 1\right) d$

Now plug in the two equation in the question given and then solve simultaneously like shown below

${a}_{3} = a + \left(3 - 1\right) d$

${a}_{5} = a + \left(5 - 1\right) d$

Therefore

$5 = a + 2 d$
$1 = a + 4 d$

$4 = - 2 d$

$d = - 2$

Now plug one of the equations given in the question as you have already found out $d$

${U}_{3} = a + \left(3 - 1\right) \left(- 2\right)$

$5 = a + 2 \left(- 2\right)$

$5 = a - 4$

$5 + 4 = a$

$a = 9$