Question

If the first and (2n-1)th terms of an AP,GP HP are equal and their nth term are a,bc respectively, then

1) a=b=c 2) a+c = b 3) abc and
ac-b2=0 4none of these

Answer 1

• Let us assume,

• first term of A.P=x

• and as given, let (2n-1)th term= y

⇒d = common difference

⇒aₙ=x+(n-1)d

⇒a₂ₙ₋₁=x+((2n-1)-1)d

• y=x+(2n-2)d

• y=x+2(n-1)d

• d=(y-x)+2(n-1)

→aₙ=a

• So,

x+(n-1)((y-x)/2(n-1)) = a

• By solving above equation,i.e, by cancelling n-1 term,

⇒x+((y-x)/2)

⇒(2x+y-x)/2

⇒a=(x+y)/2

G.P:  

1st term=x,   Common ratio=r

• aₙ=arⁿ⁻¹

• a₂ₙ₋₁xr²ⁿ⁻²

• y/x = r²ⁿ⁻²

• r=(y/x)∧1/2(n-1)

by solving we get b=(xy)∧0.5

HP:

1/y - 1/x = 2(n-1)d

we get by solving, c= 2xy/x+y

a-b=(x+y)/2-√xy

a>=b  

by multiplying values,we get a*c=b²

so,a/b>=1

∴a>b>c and ac-b²=0