The sum of the first three numbers of an arithmetic series is 12. If the 20th term -32, what is first term and the common difference?

2 Answers
Tony B
May 11, 2018

First term is 6
Common difference is -2

Explanation:

#color(blue)("Building the structure")#

The fact that the #20^("th")# term is negative tells us that the common difference is negative.

Let the common difference be #d#
Let the first term be #a#

#a_1=a_1-(0d)#
#a_2=a_1-(1d)#
#a_3=a_1-(2d)#

From this we have #a_n=a_1-(n-1)d" "..................Equation(1)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Solving for "a_1 and d)#

#color(brown)("The first three terms condition")#

#a_1=a_1-(0d)#
#a_2=a_1-(1d)#
#ul(a_3=a_1-(2d) larr" Add")#
#color(white)("dddd")3a_1-3d=12" ".................Equation(2)#

#color(brown)("Dealing with the 20th term")#

From #Eqn(1)# #a_20=a_1-(19d)=-32#

#a_1-19d=-32" "........................Equation(3)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(brown)("We now have simultaneous equations.")#

#3a_1-3d=12" ".................Equation(2)#
#a_1-19d=-32" "..............Equation(3)#

#Eqn(2)-3Eqn(3)#

#+54d=108#

#color(blue)(d=2)" "...................................Equation(4)#

Using #Eqn(4)# substitute for #d# in #Eqn(2)#

#color(green)(3a_1-3color(red)(d)=12 color(white)("dddd") ->color(white)("dddd") 3a_1-3(color(red)(2))=12)#

#color(green)(color(white)("ddddddddddddddd") ->color(white)("ddddd") a_1=(12+6)/3)#

#color(blue)(a_1=6)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Check")#

#a_20->6-(19xx2)=-32#

#a_20->-32=-32 color(red)(larr" True")#

#color(white)("d") #

#3a_1-3d=12#

#3(6)-3(2)=12#

#12=12 color(red)(larr" True")#

maganbhai P.
May 11, 2018

First term #=a=6# and common difference =#d=-2#

Explanation:

Let #a# be the first term and #d# be the common difference of arithmetic series.

So, #color(red)(n^(th) term=a_n=a+(n-1)d#

Now , the sum of first three terms is:

#a+(a+d)+(a+2d)=12#

#=>3a+3d=12#

#=>a+d=4#

#=>color(blue)(a=4-d...to(1)#

Given that, #20^(th) term=-32#

#=>a_20=a+(20-1)d=-32#

#:.color(blue)(a+19d=-32...to(2)#

Subst. #a=4-d # into #(2)#

#4-d+19d=-32#

#=>18d=-32-4=-36#

#d=-2#

Subst. #d=-2# into #(1)#

#a=4-(-2)=4-2=6#

Hence, first term #=a=6#

and common difference =#d=-2#

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