# How to find the first term, the common difference, and the nth term of the arithmetic sequence here? 8th term is 4; 18th term is -96

Jul 6, 2018

First term is $74$ , common difference is $- 10$
$n$ th term of A.P. series is ${T}_{n} = a + \left(n - 1\right) d$

#### Explanation:

Let $a , d , n$ be the first term , common difference and number of

terms of an A.P. series

$n$ th term of A.P. series is ${T}_{n} = a + \left(n - 1\right) d$

$8$ th term of A.P. series is ${T}_{8} = a + \left(8 - 1\right) d = 4$or

a + 7 d=4 ; (1)

$18$ th term of A.P. series is ${T}_{18} = a + \left(18 - 1\right) d = - 96$or

a + 17 d=-96 ; (2)  , subtracting equation (1) from equation (2)

we get , $10 d = - 100 \mathmr{and} d = - 10$ putting $d = - 10$ in

equation (1) we get, $a - 70 = 4 \mathmr{and} a = 74$

Hence first term is $74$ and common difference is $- 10$ [Ans]

Jul 6, 2018

The first term is $74$ and common difference is $- 10$

#### Explanation:

If a Arithmatic Sequence has first term $a$ and the common difference $d$ then the formula of ${n}^{t h}$ term is
${a}_{n} = a + \left(n - 1\right) d$ ......(1)

According to question
${8}^{t h}$ term is $4$
Put n=8 in equation (1)
$\implies {a}_{8} = a + \left(8 - 1\right) d = a + 7 d$
but ${a}_{8} = 4$
Hence $\implies a + 7 d = 4$ ........(2) $\implies a = 4 - 7 d$
and
${18}^{t h}$ term is $- 96$
Put n=18 in equation (1)
$\implies {a}_{18} = a + \left(18 - 1\right) d = a + 17 d$
but ${a}_{18} = - 96$
Hence $\implies a + 17 d = - 96$ ........(3)
by putting value of $a$ from equation (2) in the equation (3)
$\implies \left(4 - 7 d\right) + 17 d = - 96$
$\implies 4 + 10 d = - 96$
Transfer $4$ to the Right Hand Side
$\implies 10 d = - 96 - 4 = - 100$
Divide by 10
$\frac{10 d}{10} = - \frac{100}{10}$
$d = - 10$

Put the value of $d$ in equation (2) to get the first term of the Arithmetic sequence
By Equation (2)
$a = 4 - 7 d = 4 - 7 \left(- 10\right)$
$a = 4 + 70$
$a = 74$

Hence the first term is $74$ and common difference is $- 10$

Common difference: $d = - 10$
first term: $a = 74$
nth term: ${T}_{n} = 74 - 10 d$

#### Explanation:

8th term
${T}_{8} = 4$

18th term
${T}_{18} = - 96$

General formula for nth tern

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

$\left(n , {T}_{n}\right) = \left(8 , 4\right) \to$
$4 = a + \left(8 - 1\right) \times d$
Simplifying
$a + 7 d = 4 - - - - - \left(1\right)$

$\left(n , {T}_{n}\right) = \left(18 , - 96\right) \to$
$- 96 = a + \left(18 - 1\right) \times d$
Simplifying
$a + 17 d = - 96 - - - \left(2\right)$

Subtrtacting (1) from (2)
$10 d = - 100$

$d = - 10$
Substituting
$d = - 10$
in (1)
$a + 7 \times \left(- 10\right) = 4$
$a - 70 = 4$
$a = 74$

${T}_{n} = 74 - 10 d$