Question 692570


An arithmetic series has a third term of 0. The sum of the first 15 terms is – 300. What is the first term?


Formula for d, or the common difference in a series: {{{d = (a[n] - a[1])/(n - 1)}}}, with {{{a[n]}}} = {{{a[3]}}} = 0, and n = 3


Therefore, {{{d = (a[n] - a[1])/(n - 1)}}} becomes: {{{d = (0 - a[1])/(3 - 1)}}} ------ {{{d = (- a[1])/2}}} ------ eq (i)


Formula for sum of a series of numbers, or {{{S[n]}}}: {{{S[n] = (n/2)(2a[1] + (n - 1)d)}}}, with {{{S[n]}}} = {{{S[15]}}} = - 300 ; and n = 15


Therefore, {{{S[n] = (n/2)(2a[1] + (n - 1)d)}}} becomes: {{{S[15] = (15/2)(2a[1] + (15 - 1)d)}}} --- {{{- 300 = 7.5(2a[1] + 14d)}}} --- {{{- 300 = 15a[1] + 105d}}} --- eq (ii)


{{{- 300 = 15a[1] + 105((- a[1])/2)}}} ------ Substituting {{{(- a[1])/2}}} for d in eq (ii)


{{{- 300 = 15a[1] + 52.5(- a[1])}}}


{{{- 300 = 15a[1] - 52.5a[1]}}}


{{{- 37.5a[1] = - 300}}}


{{{a[1]}}}, or 1st term = {{{(- 300)/- 37.5}}}, or {{{highlight_green(8)}}}


You can do the check!!


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