1 1^ 2^ + 1 2^ 3^ + + 1 89^ + 90^ =

Step-by-step solution

Here, 1 1^ 2^ + 1 2^ 3^ + + 1 89^ 90^ = 1 1^ [ array l (2^ -1^ ) 1^ 2^ + (3^ -2^ ) 2^ 3^ + . . + (90^ -89^ ) 89^ 90^ array ] We know that (A-B)= A B- A B aligned = 1 l^ [ array l 2^ 1^ - 2^ l^ 1^ 2^ + 3^ 2^ - 3^ 2^ 3^ 2^ + . .+ 90^ 89^ - 90^ 89^ 89^ 90^ array ] aligned array r = 1 1^ [ 1^ - 2^ + 2^ - 3^ + . . . + 89^ - 90^ ] array = 1 1^ [ 1^ - 90^ ]= 1^ 1^ = 1^ ^2 1^