Question

9^ log3 (1+0,5+0,25...)

Answer 1

1+0,5+0,25+... - геометрическая прогрессия ее сумма равна 2 (1/(1-0,5=2).

log₃(2) - такое число, что 3 в его степени равно 2, поэтому 9 в степени log₃(2)=4

Ответ: 4

Answer 2

Let's break down the expression step by step:

1. First, let's look at the infinite geometric series inside the logarithm: 1 + 0.5 + 0.25 + ...

This is a geometric series with a common ratio of 0.5 (each term is half of the previous one) and a first term of 1. The sum of an infinite geometric series is given by the formula:

Sum = a / (1 - r)

Where: a = first term = 1 r = common ratio = 0.5

Sum = 1 / (1 - 0.5) Sum = 1 / 0.5 Sum = 2

So, the infinite geometric series inside the logarithm sums to 2:

log3(2)

2. Now we have:

9^log3(2)

This is equivalent to:

3^2^log3(2)

Now, we can simplify further:

log3(2) is the exponent we raise 3 to in order to get 2.

So, the expression simplifies to:

3^(2^log3(2))

Since log3(2) is the exponent we raise 3 to in order to get 2, this can be rewritten as:

3^(2^log3(2)) = 3^2

And finally:

3^2 = 9

So, the value of the expression is 9.