Question

Прогрессия1/2+2/3+3/2+4/3+... +15/2+16/3

Answer 1

1/2+2/3=3+4/6=7/6;3/2+4/3=9+8/=17/6;15/2+16/3=45+32/6=77/6,это мы все к общему знаменателю привели,теперь все дроби с одинаковым знаменателем нужно сложить :7/6+17/6+77/6=101/6=16,8

Answer 2

The given expression represents a series with a pattern of fractions. Let's break it down step by step:

1/2 + 2/3 + 3/2 + 4/3 + ... + 15/2 + 16/3

First, let's observe the pattern:

The numerators of the fractions are in ascending order from 1 to 16, and the denominators alternate between 2 and 3.

To find the sum of this series, we need to determine its value. We can do this by finding a common denominator for all the fractions and then adding them together.

The least common multiple (LCM) of 2 and 3 is 6. So, we'll convert all the fractions to have a denominator of 6:

1/2 = (1/2) * (3/3) = 3/6 2/3 = (2/3) * (2/2) = 4/6 3/2 = (3/2) * (3/3) = 9/6 4/3 = (4/3) * (2/2) = 8/6 ... 15/2 = (15/2) * (3/3) = 45/6 16/3 = (16/3) * (2/2) = 32/6

Now, the series becomes:

3/6 + 4/6 + 9/6 + 8/6 + ... + 45/6 + 32/6

Next, add all the fractions together:

(3 + 4 + 9 + 8 + ... + 45 + 32) / 6

Now, calculate the sum of the numerators:

(3 + 4 + 9 + 8 + ... + 45 + 32) = (1 + 2 + 3 + 4 + ... + 15 + 16) = sum of the first 16 natural numbers.

The sum of the first n natural numbers is given by the formula: sum = n * (n + 1) / 2

So, the sum of the numerators = 16 * (16 + 1) / 2 = 16 * 17 / 2 = 136

Now, put it back into the original expression:

Sum = 136 / 6 = 68 / 3

Therefore, the sum of the given series is 68/3.