Math, asked by sreenu204308, 11 months ago

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​

Answers

Answered by IamIronMan0
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}

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