Question

1/15 + 1/35 + 1/63 + 1/99 +... +1/ 255​

Answer 1

Ответ:

7/51

Пошаговое объяснение:

Здесь воспользуемся приёмом представления дробей в виде разности.

1/15 +1/35 +1/63 +1/99 +...+1/255=1/2 ·(1/3 -1/5) +1/2 ·(1/5 -1/7) +1/2 ·(1/7 -1/9) +1/2 ·(1/9 -1/11) +1/2 ·(1/11 -1/13) +1/2 ·(1/13 -1/15) +1/2 ·(1/15 -1/17)=1/2 ·(1/3 -1/5 +1/5 -1/7 +1/7 -1/9 +1/9 -1/11 +1/11 -1/13 +1/13 -1/15 +1/15 -1/17)=1/2 ·(1/3 -1/17)=1/2 ·(17-3)/51=1/2 ·14/51=7/51

Answer 2

To find the sum of the given series:

1/15 + 1/35 + 1/63 + 1/99 + ... + 1/255

You can see that the denominators follow a pattern. The denominators are all multiples of consecutive odd numbers. Let's try to express the pattern:

Denominator for the first term: 15 (3 * 5) Denominator for the second term: 35 (5 * 7) Denominator for the third term: 63 (7 * 9) Denominator for the fourth term: 99 (9 * 11) ...

The common difference between the odd numbers is 2. So, the nth term of the series can be expressed as:

Denominator for the nth term = (2n + 1)(2n + 3)

Now, let's write the series with these denominators:

1/[(2 * 1 + 1)(2 * 1 + 3)] + 1/[(2 * 2 + 1)(2 * 2 + 3)] + 1/[(2 * 3 + 1)(2 * 3 + 3)] + ...

1/[(3 * 5)] + 1/[(5 * 7)] + 1/[(7 * 9)] + ...

Now, the series looks like:

1/(15) + 1/(35) + 1/(63) + ...

To find the sum of this series, you can use the following formula for the sum of an infinite geometric series:

Sum = a / (1 - r)

In this case, a = 1/15 (the first term) and r is the common ratio between terms, which is 35/15 = 7/3.

Now, plug these values into the formula:

Sum = (1/15) / (1 - 7/3)

To simplify this:

Sum = (1/15) / (3/3 - 7/3) Sum = (1/15) / (-4/3)

Now, divide 1/15 by -4/3:

Sum = (1/15) * (-3/4) Sum = -3/60 Sum = -1/20

So, the sum of the series is -1/20.