Question 934991
 find {{{a[1]}}} and the common ratio({{{r}}})if given {{{a[n]=1}}}, {{{n=7}}}, {{{s[n]=1}}}

we know that

{{{a[n]=a[1]+(n-1)d}}}...plug in given values

{{{1=a[1]+(7-1)d}}}

{{{1=a[1]+6d}}}...solve for {{{d}}} in terms of {{{a[1]}}}

{{{d=(1-a[1])/6}}}...........eq.1


we also know that the sum is:

{{{s[n]=(n/2)(a[1] +a[n])}}}...plug in given values

{{{1=(7/2)(a[1] +1)}}}.........solve for {{{a[1]}}}

{{{2=7(a[1] +1)}}}

{{{2=7a[1] +7}}}

{{{2-7=7a[1] }}}

{{{-5=7a[1] }}}

{{{highlight(a[1] =-5/7)}}}

go to {{{d=(1-a[1])/6}}}.....eq.1 plug in {{{a[1]}}}

{{{d=(1-(-5/7))/6}}}

{{{d=(1+5/7)/6}}}

{{{d=(12/7)/6}}}

{{{d=12/42}}}....reduce, divide both numerator and denominator by {{{6}}}

{{{highlight(d=2/7)}}}


check:

{{{1=-5/7+(7-1)(2/7)}}}

{{{1=-5/7+(6)(2/7)}}}

{{{1=-5/7+12/7}}}

{{{1=7/7}}}

{{{1=1}}}