First, there are two formulas for working with arithmetic sequences and series that will be helpful:
#a_n = a_1 + (n-1)d color(white)("aaaaa")"General Term Formula"#
#sum_{i=1}^n a_i = (a_1+a_n)(n/2) color(white)("aaaaa") "Sum Formula"#
Now, we know there will be five parts, so #n=5#. We also know that the ratio of the first and last parts will be 2:3. Thus:
#a_1/a_5 = 2/3 => a_1 = 2/3a_5#
Lastly, since we know the value 12.5 will be the sum of the five terms, we can use the sum formula to express that sum of 12.5 in terms of #a_1# and #a_5#, and then substitute this previous fact into it:
#Sum = (a_1+a_n)(n/2)#
#12.5 = (a_1+a_5)(5/2)#
#12.5 = (2/3a_5 + a_5)(5/2)#
#12.5(2/5) = (2/3a_5 + a_5)#
#5 = 5/3a_5#
#3 = a_5#
#:. a_1 = 2/3(3) = 2#
We can now determine the common difference between the terms using the general term formula, which will let us write out the five term sequence:
#a_n = a_1 + (n-1)d #
#a_5 = a_1 + (5-1)d #
#3 = 2 + 4d #
#1 = 4d => d = 1/4#
Thus, the five terms are #2, 2.25, 2.5, 2.75, and 3#.
Check:
#2 + 2.25 + 2.5 + 2.75 + 3 = 12.5#
