1) The value of x+y+z=15, if a,x,y,z,b are in A.P while the vlue of 1/x + 1/y + 1/z =5/3 if a,x,y,z,b are in H.P. , Find a and b.
2) If the (m+!)th,(n+1)th and (r+1)th terms of an A.P are in G.P. and m,n,r are in H.P. Then show that the ratio of the first term to the common difference of the A.P is -n/2.
3) Let a,b,c are respectively the sums of the first n terms, the next n terms, and the next n terms of a G.P. Show that a,b,c are in G.P.
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3 Answers
11)
when a,x,y,z,b are in AP
x+y+z = 15
3y= 15 ( 2y=x+z)
y=5
now a+b/2 = y
a+b = 2*5= 10
now if a,x,y,z,b are in H.P
1/x+1/y+1/z= 5/3
3/y = 5/3 ( since 2/y = 1/x+1/z)
y= 9/5
again
1/a+1/b = 2/y = 10/9
a+b/ab = 10/9
ab = 9 ( since a+b = 10)
solving for a and b
a= 9,1
b= 1,9
622) If the (m+1)th,(n+1)th and (r+1)th terms of an A.P are in G.P. and m,n,r are in H.P. Then show that the ratio of the first term to the common difference of the A.P is -n/2.
1/m+1/r=2/n
(a+md)(a+rd)=(a+nd)2
a2+(m+r)ad+mrd2=a2+n2d2+2and
(m+r-2n)a/d+mr-n2=0
(1/m+1/r-2n/mr)a/d+1-n2/mr=0
2/n(a/d)(1-n2/mr)+(1-n2/mr)=0
thus, a/d=-n/2
13)
Sum of first n terms = a(rn-1)/r-1
sum of next n terms = a(r2n-1)/r-1 - a(rn-1)/r-1
= {a(rn-1)/r-1} { rn - 1 +1 )
= {rn*(a(rn-1)} /r-1
sum of next n terms = a(r3n-1)/(r-1) - a(r2n-1)/r-1
= a(rn-1)/r-1 { r2n+1 +rn -rn - 1)
= (r2n){a(rn-1)/r-1 }
thus we see that the sums for an G.P with
first term as = a(rn-1)/r-1
and common ratio = rn
