#color(blue)("Consider the Arithmetic version")#
#a_1=a_1+0d = 3color(white)("d")........# where #d# is the common difference
#a_2=a_1+0d+d#
#a_3=a_1+0d+d+d#
#a_4=a_1+0d+d+d+d #
#a_5=a_1+0d+d+d+d+d -> a_1+4d#
So for any #n# we have: #color(red)(a_n=3+(n-1)d)#
Given that #a_5=3+4d=48#
#color(white)("dddd")=>color(red)(color(white)("d")d=(48-3)/4 = 45/4)#
#a_1=3 color(white)("d")larr" Given"#
#a_2=3+45/4(2-1) = 3+(45/4xx1) = 57/4#
#a_3=3+45/4(3-1) = 3+(45/4xx2) = 51/2 #
#a_4=3+45/4(4-1)=3+(45/4xx3) = 147/4#
#a_5 = 3+45/4(5-1)=3+(45/4xx4) = 48 larr" As given"#
For AP #a=57/4; color(white)("d")b= 51/2; color(white)("d")c=147/4#
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#color(blue)("Consider the Geometric version")#
#a_1=a_1r^0=3#
#a_2=a_1r#
#a_3=a_1r^2#
#a_4=a_1r^3#
#a_5=a_1r^4=48#
General rule: #a_n=a_1r^(n-1)#
Given that #a_1=3# then #a_5=3r^4=48#
#r^4=48/3 = 16#
#r=root(4)(16) = 2#
Thus
#a_2=a=3(2)^(2-1)=6#
#a_3=b=3(2)^(3-1)=12#
#a_4=c=3(2)^(4-1)=24#
For GP #a=6;color(white)("d")b=12:color(white)("d")c=24#
