x₁ = -2

x₂ = x₁ + 4(1) - 2 = -2 + 4 - 2 = 0

x₃ = x₂ + 4(2) - 2 = 0 + 8 - 2 = 6

x₄ = x₃ + 4(3) - 2 = 6 + 12 - 2 = 16

The sequence diverges to infinity. The increase from one term to the next is always 4 more than the previous increase.

We have to first rewrite the equation as :

xₙ₊₁ - xₙ = 4n - 2

Then,

xₙ

= (xₙ - xₙ₋₁)
+ (xₙ₋₁ - Xₙ₋₂)
+ (xₙ₋₂ - xₙ₋₃)
+ ...
+ ...
+ (x₃ - x₂)
+ (x₂ - x₁)
+ x₁


= 4(n - 1) - 2
+ 4(n - 2) - 2
+ 4(n - 3) - 2
+...
+...
+ 4(2) - 2
+ 4(1) - 2
-2


= 4(n - 1 + n - 2 + n - 3 + ... + 2 + 1) - 2n

= 4(n - 1 + 1)(n -1)/2 - 2n

The 4 can be factored out, such tht the term in the brackets are Gauss-paired.

i.e sum of an A.P,
where number of terms = n - 1
First term = 1, last term = n - 1
Common difference = 1

There are also n times of -2 so it's grouped together at the back as -2n.


= 2n(n - 1) - 2n
= 2n(n - 1 - 1)
= 2n(n - 2)

(Shown)

Lastly,


xₙ / n²

= 2n(n - 2) / n²

= (2n² - 4n) / n²

= 2 - 4/n


For large values of n, 4/n is very small.

4/n → 0 as n → ∞

So 2 - 4/n → 2

The xₙ/n² converges to 2.

Alternative presentation to prove xₙ if you're not required to use method of difference:

xₙ

= xₙ₋₁ + 4(n - 1) - 2

= xₙ₋₂ + 4(n - 2) - 2 + 4(n - 1) - 2

= xₙ₋₃ + 4(n - 3) - 2 + 4(n - 2) - 2 + 4(n - 1) - 2

= ...

= x₁ + 4(1) - 2 + 4(2) - 2 + 4(3) - 2 + ... + 4(n - 1) - 2

= -2 + 4(1+2+3+...+n-1) - 2(n - 1)

= 4(1 + n - 1)(n-1)/2 - 2(n - 1 + 1)

= 2n(n - 1) - 2n

= 2n(n - 1 - 1)

= 2n(n - 2)

thank youuu :)))