Question

The sequence a1,a2,a3,... satisfies a1 =19, a9 =99, and,for all n≥3,an is the arithmetic mean of the first n − 1 terms. Find a2.

Answer 1

Answer:

179

Step-by-step explanation:

Hi,

Given a₁, a₂, a₃,........ a sequence in which a₁ = 19 and a₉ = 99

Now let a₂ = x.

Given ∀n ≥ 3, an is the arithmetic mean of the first n-1 terms

⇒ a₃ = (a₁ + x)/2

a₄ = (a₁ + a₂ + a₃)/3

    =(a₁ + x + (a₁ + x)/2)/3

   = (a₁ + x)/2

So on, ∀n ≥ 3, we get an =  (a₁ + x)/2

But given a₉ = 99

⇒ (19+ x)/2 = 99

⇒x = 179

Thus, the value of a₂ is 179.

Hope, it helped !