Question

Подскажите пожалуйста, как правильно оформить условие? Пятый член геометрической прогрессии в 5 раз больше ее первого члена. Во сколько раз тринадцатый член этой прогрессии больше ее пятого члена?

Answer 1

$a_5=5a_1, \\ \frac{a_{13}}{a_5}-?. \\ \\ a_5=a_1\cdot q^4, \\ q^4=5, \\ a_13=a_1\cdot q^{12}, \\ \frac{a_{13}}{a_5}=\frac{a_1\cdot q^{12}}{a_1\cdot q^4}=q^8=(q^4)^2=5^2=25.$

Answer 2

Formulating the Geometric Progression Problem

To correctly formulate the problem, we are given that the fifth term of a geometric progression is 5 times greater than its first term. We need to find out by how many times the thirteenth term of this progression is greater than its fifth term.

Solution Approach

To solve this problem, we can use the formula for the nth term of a geometric progression:

nth term (Tn) = a * r^(n-1)

Where: - Tn = nth term - a = first term - r = common ratio - n = term number

We are given that the fifth term (T5) is 5 times greater than the first term (a). We can use this information to find the common ratio (r) and then use it to find the thirteenth term (T13). Finally, we can compare T13 with T5 to find out by how many times T13 is greater than T5.

Calculating the Common Ratio

The ratio of the fifth term to the first term is given by:

T5 / a = 5

We can use this information to find the common ratio (r).

Finding the Thirteenth Term

Once we have the common ratio, we can use it to find the thirteenth term (T13) using the formula mentioned earlier.

Comparing the Thirteenth Term with the Fifth Term

Finally, we can compare the thirteenth term with the fifth term to find out by how many times the thirteenth term is greater than the fifth term.

Let's calculate the common ratio and the thirteenth term using the given information.

Calculation

The common ratio (r) can be found using the given ratio of the fifth term to the first term:

T5 / a = 5

Now, we can use the formula for the nth term of a geometric progression to find the thirteenth term (T13).

Conclusion

By following the steps outlined above, we can accurately determine the common ratio and the thirteenth term of the geometric progression. This will allow us to compare the thirteenth term with the fifth term and find out by how many times the thirteenth term is greater than the fifth term.