progressions

3. The nth term = T_n = n(n) !

T_n can be written as

T_n = (n + 1 – 1) (n)!

T_n = (n + 1) ! – (n) ! ..........(i)

This is in the form f(n) – f(n – 1)

S = T₁ + T₂ + T₃ + T₄ + .......... + T_n

S = (2! – 1!) + (3! – 2!) + (4! – 3!) + ......... + {(n + 1)! – n!}
S = – 1! + (n + 1)!

→ S = (n + 1)! – 1

2.

The differences of successive terms are in G.P.

Let S = sum of first n terms.

⇒ S = 1 + 4 + 10 + 22 + 46 + .......... + T_n

⇒ S = 1 + 4 + 10 + 22 + 46 + ............ + T_n–1 + T_n

On subtracting, we get
0 = 1 + {3 + 6 + 12 + 24 + ............} – T_n

T_n = 1 + {3 + 6 + 12 + 24 + ........(n – 1) terms}

now you can solve

1)

(a+b+c+d+...)^2 - a^2 + b^2 + c^2 + d^2 ... = ??