Question

let an be the units place of 1^2+2^2+3^2+.........+n^2 prove that the decimal 0.a1a2a3.....an is a rational number and represent it has P/Q form. where p and Q are natural numbers​

Answer 1

Answer:

Unit places will repeat a pattern after 10

From 1 to 10

1 , 4 , 9 , 6 , 5 , 4 , 9 , 4 , 1 , 0 , 1 , 4.....

Now your decimal of course can be written a P/Q

Form just remove decimal and divide 10^n .

Now let

$n \: \to \: \infty$

$x = 0.1496549410...1496... \\ {10}^{10} x = 1496549410.1496549410..... \\ {10}^{10} x = 1496549410 + 0.1496549410...... \\ {10}^{10} x = 1496549410 + x \\ {10}^{10} x - x = 1496549410 \\ 999999999x = 1496549410 \\ \\ x \: = \frac{1496549410 }{999999999}$