Question

a series in a.p has a sum of 120.if the 3rd term is 7 and it's 7th term is 15.find the number of terms in the series.

Answer 1

So you have that

ar+ar2=12ar2+ar3=60" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; font-size: 15px; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">ar+ar2=12ar2+ar3=60ar+ar2=12ar2+ar3=60

Then

a=12r+r2ar2+ar3=60" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; font-size: 15px; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">a=12r+r2ar2+ar3=60a=12r+r2ar2+ar3=60

⟹" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; font-size: 15px; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⟹⟹

a=12r+r212r+r2r2+12r+r2r3=60" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; font-size: 15px; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">a=12r+r212r+r2r2+12r+r2r3=60a=12r+r212r+r2r2+12r+r2r3=60

⟹" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; font-size: 15px; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⟹⟹

12r+r2r2+12r+r2r3=6012r+r2r+r212" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; font-size: 15px; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">12r+r2r2+12r

Answer 2

Sum=120
A+2d=7
A+6d=15
-4d=8
D=-2
A=7+4
A=11
120=n/2(2a+(n-1)d)
120=n/2(22+(n-1)-2)
240=22n-2n^2+2n
2n^2-20n+240=0
N^2-10n+120=0
(N-40)(n+30)=0
N=40
Rejecting negative values...