Question

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

Answer 1

First term is `74` , common difference is `-10`
`n` th term of A.P. series is `T_n= a +(n-1)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 +(n-1)d`

`8` th term of A.P. series is `T_8= a +(8-1)d=4`or

`a + 7 d=4 ; (1)`

`18` th term of A.P. series is `T_18= a +(18-1)d=-96`or

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

we get , `10 d= -100 or d = -10` putting `d=-10` in

equation (1) we get, `a - 70 =4 or a = 74`

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

Answer 2

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^(th)` term is
`a_n=a+(n-1)d` ......(1)

According to question
`8^(th)` term is `4`
Put n=8 in equation (1)
`=> a_8=a+(8-1)d=a+7d`
but `a_8=4`
Hence `=>a+7d=4` ........(2) `=> a=4-7d`
and
`18^(th)` term is `-96`
Put n=18 in equation (1)
`=> a_18=a+(18-1)d=a+17d`
but `a_18=-96`
Hence `=>a+17d=-96` ........(3)
by putting value of `a` from equation (2) in the equation (3)
`=>(4-7d)+17d=-96`
`=>4+10d=-96`
Transfer `4` to the Right Hand Side
`=>10d=-96-4=-100`
Divide by 10
`(10d)/10=-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=4-7d=4-7(-10)`
`a=4+70`
`a=74`

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

Answer 3

Common difference: `d=-10`
first term: `a=74`
nth term: `T_n=74-10d`

Explanation:

8th term
`T_8=4`

18th term
`T_18=-96`

General formula for nth tern

`T_n=a+(n-1)d`

`(n,T_n)=(8,4)->`
`4=a+(8-1)xxd`
Simplifying
`a+7d=4-----(1)`

`(n,T_n)=(18,-96)->`
`-96=a+(18-1)xxd`
Simplifying
`a+17d=-96---(2)`

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

`d=-10`
Substituting
`d=-10`
in (1)
`a+7xx(-10)=4`
`a-70=4`
`a=74`

`T_n=74-10d`