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

3 Answers
Jul 6, 2018

First term is 7474 , common difference is -1010
nn th term of A.P. series is T_n= a +(n-1)dTn=a+(n1)d

Explanation:

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

terms of an A.P. series

nn th term of A.P. series is T_n= a +(n-1)dTn=a+(n1)d

88 th term of A.P. series is T_8= a +(8-1)d=4 T8=a+(81)d=4or

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

1818 th term of A.P. series is T_18= a +(18-1)d=-96 T18=a+(181)d=96or

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

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

equation (1) we get, a - 70 =4 or a = 74a70=4ora=74

Hence first term is 7474 and common difference is -1010 [Ans]

Jul 6, 2018

The first term is 7474 and common difference is -1010

Explanation:

If a Arithmatic Sequence has first term aa and the common difference dd then the formula of n^(th)nth term is
a_n=a+(n-1)dan=a+(n1)d ......(1)

According to question
8^(th)8th term is 44
Put n=8 in equation (1)
=> a_8=a+(8-1)d=a+7da8=a+(81)d=a+7d
but a_8=4a8=4
Hence =>a+7d=4a+7d=4 ........(2) => a=4-7da=47d
and
18^(th)18th term is -9696
Put n=18 in equation (1)
=> a_18=a+(18-1)d=a+17da18=a+(181)d=a+17d
but a_18=-96a18=96
Hence =>a+17d=-96a+17d=96 ........(3)
by putting value of aa from equation (2) in the equation (3)
=>(4-7d)+17d=-96(47d)+17d=96
=>4+10d=-964+10d=96
Transfer 44 to the Right Hand Side
=>10d=-96-4=-10010d=964=100
Divide by 10
(10d)/10=-100/1010d10=10010
d=-10d=10

Put the value of dd in equation (2) to get the first term of the Arithmetic sequence
By Equation (2)
a=4-7d=4-7(-10)a=47d=47(10)
a=4+70a=4+70
a=74a=74

Hence the first term is 7474 and common difference is -1010

Common difference: d=-10d=10
first term: a=74a=74
nth term: T_n=74-10dTn=7410d

Explanation:

8th term
T_8=4T8=4

18th term
T_18=-96T18=96

General formula for nth tern

T_n=a+(n-1)dTn=a+(n1)d

(n,T_n)=(8,4)->(n,Tn)=(8,4)
4=a+(8-1)xxd4=a+(81)×d
Simplifying
a+7d=4-----(1)a+7d=4(1)

(n,T_n)=(18,-96)->(n,Tn)=(18,96)
-96=a+(18-1)xxd96=a+(181)×d
Simplifying
a+17d=-96---(2)a+17d=96(2)

Subtrtacting (1) from (2)
10d=-10010d=100

d=-10d=10
Substituting
d=-10d=10
in (1)
a+7xx(-10)=4a+7×(10)=4
a-70=4a70=4
a=74a=74

T_n=74-10dTn=7410d