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

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Answer 1 · 2018-03-22T08:08:15

#"First Term " a_1 = 9, " Common Difference " d = -2#

Given #a_3 = 5, a_5 = 1, " To find First Term " a_1, " common difference " d#

#a_3 = a_1 + (3 - 1) * d#

#a_1 + 2 d = 5, " Eqn 1 "#

#a_5 = a_1 + (5-1) * d #

#a_1 + 4d = 1, " Eqn 2 "#

Solving equations (1), (2)

#" Eqn 2 - Eqn 1 "#

#4d - 2d = 1 - 5 = -4#

#2 d = -4 " or " d = -2#

Substituting value of d in Eqn (1),

#a_1 + 2 * -2 = 5#

#a_1 = 5 + 4 = 9#

Answer 2 · 2018-03-22T08:14:29

#a_1= 9# and #d=-2#.

#U_n= a+(n-1)d#

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

#a_3= a+(3-1)d#

#a_5= a+(5-1)d#

Therefore

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

#4= -2d#

#d=-2 #

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

#U_3=a+(3-1)(-2)#

#5=a+2(-2)#

#5=a -4#

#5+4=a#

#a=9#