Question

Three numbers are in ap and their sum is 15. If 1,3,9, be added to them respectively , they form a gp. Find the numbers

Answer 1

Answer:

3,5,7and 15,5,-5

Step-by-step explanation:

Let numbers in ap be (a-d),a,(a+d)

ATQ , sum of numbers =15

(a-d)+a+(a+d)=15

-d and +d cancel out each other so we get

3a =15

a=15/3=5

So now terms are

(5-d), 5 ,(5+d)

Now ATQ,1,3,9 are added to first ,second and third term so terms becomes

(5-d)+1,(5+3),(5+d)+9 ie (6-d),8,(14+d)

Now ATQ, these new terms are in GP

so (6-d)(14+d) =8²

=> 84+6d-14d-d² =64

=> -d²-8d+84-64 =0

=> -d²-8d +20=0

=> d²+8d-20=0

now we factrioze it by splitting the middle term

we have to factrioze 20 in to two factors whose product are 20 and whose difference is 8

these two factors are 10 and 2 now

=> d²+(10-2)d-20=0

=> d²+10d-2d-20=0

=> d(d+10)-2(d+10)=0

=> (d+10)(d-2)=0

now taking d-2=0 =>d=2

Now numbers are

a -d=5-2=3 ,a=5,a+d=5+2=7

So ans is 3,5,7

If d+10=0=> d =-10

Now numbers are

a-d=5-(-10)=5+10=15

a=5

a+d=5+(-10)=5-10=-5

So numbers are 15 ,5,-5