Question

Найти значение а+а^2+...+а^2020/ а^-1+а^-2+...а^-2020

Answer 1

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Answer 2

To find the value of the expression (a + a^2 + ... + a^2020) / (a^-1 + a^-2 + ... + a^-2020), we can use the formula for the sum of a geometric series.

The sum of a geometric series with the first term "a" and the common ratio "r" is given by:

S = a * (1 - r^n) / (1 - r),

where "n" is the number of terms in the series.

In our case, we have two geometric series:

1. The sum of a + a^2 + ... + a^2020: a = a (first term) r = a (common ratio, each term is multiplied by "a" to get the next term) n = 2020 (number of terms)

2. The sum of a^-1 + a^-2 + ... + a^-2020: a = 1/a (first term, the reciprocal of "a") r = 1/a (common ratio, each term is multiplied by 1/a to get the next term) n = 2020 (number of terms)

Now, let's calculate both sums:

1. Sum of a + a^2 + ... + a^2020: S1 = a * (1 - a^2020) / (1 - a).

2. Sum of a^-1 + a^-2 + ... + a^-2020: S2 = (1/a) * (1 - (1/a)^2020) / (1 - 1/a).

Finally, we can find the value of the expression by dividing S1 by S2:

(a + a^2 + ... + a^2020) / (a^-1 + a^-2 + ... + a^-2020) = S1 / S2 = (a * (1 - a^2020) / (1 - a)) / ((1/a) * (1 - (1/a)^2020) / (1 - 1/a)) = (a * (1 - a^2020) / (1 - a)) / ((1 - (1/a)^2020) / ((1 - 1/a) * a)) = (a * (1 - a^2020) / (1 - a)) * ((1 - a) / (1 - (1/a)^2020)) = a * (1 - a^2020) * ((1 - a) / (1 - (1/a)^2020)).

The expression is now simplified, and if you plug in a specific value for "a," you can find its numerical value.