Three numbers are in the ratio 1:4:12. if 1 is added to the first number,the resulting number together with other two numbers from a GP.find the original three numbers.
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11th
Maths
Sequences and Series
Introduction to Geometric Progression
In a GP the sum of three nu...
MATHS
In a GP the sum of three numbers is 14, if 1 is added to first two numbers and the third number is decreased by 1, the series becomes AP, find the geometric sequence.
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ANSWER
Letthenumbersana,ar,ar
2
wherea=firsttermandr=commonratio
∴a+ar+ar
2
=14
and
a+1,ar+1,ar
2
−1areinAP
∴2(ar+1)=(a+1)+(ar
2
−1)
or2ar+2=a+ar
2
or3ar+2=a+ar+ar
2
or3ar+2=14
∴ar=4
ora=(
r
4
)
∴(
r
4
)+(
r
4
)×r+(
r
4
)×r
2
=14
or(
r
4
)+4+4r=14
or4+4r+4r
2
=14r
or4r
2
−10r+4=0
or4r
2
−8r−2r+4=0
or4r(r−2)−2(r−2)=0
or(4r−2)(r−2)=0
∴r=(
2
1
)or2
ifr=(
2
1
),thena=(
(
2
1
)
4
)=8
∴sequence8,4,2
andifr=2,thena=(
2
4
)=2
∴sequence2,4,8