1)Prove that if n is an odd positive integer, then
N = 2269n + 1779n + 1730n ¡ 1776n
is an integer multiple of 2001... 2)Factor the expression
30(a2 + b2 + c2 + d2) + 68ab ¡ 75ac ¡ 156ad ¡ 61bc ¡ 100bd + 87cd: 3) Prove that for every x∈R is true this equation: x^8+x^6-x^3-x+1>0,,, 4) Make the add:1^2+2^2-3^2+… +〖( n-1)〗^2+n^2 which n∈ N ,,,,5) Find the last digit of 777^777
Some of the problems are impossible to read.
There is a formula for this series: .4) Find this sum: . where
But I suppose they want to see its derivation . . .
We are concerned with the last digit (only) of any number.5) Find the last digit of
Note the endings of consecutive powers of 7.
We see that ends in 1 ... and the sequence repeats.
We have: .
Therefore, the last digit of is