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 #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]

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^(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#

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#