Question

Найдите сумму первых пяти членов геометрической прогрессии, третий член которой равен 3 , а пятый 27. ПОМОГИТЕ!!

Answer 1

Calculation of the Sum of the First Five Terms of a Geometric Progression

To find the sum of the first five terms of a geometric progression, we need to know the third term and the fifth term of the progression. In this case, the third term is given as 3 and the fifth term is given as 27.

Let's denote the first term of the geometric progression as a and the common ratio as r. The formula to find the sum of the first n terms of a geometric progression is:

S_n = a * (r^n - 1) / (r - 1)

We can use this formula to find the sum of the first five terms.

Solution

Given: - Third term (a_3) = 3 - Fifth term (a_5) = 27

We can use the formula for the nth term of a geometric progression to find the common ratio (r):

a_n = a * r^(n-1)

Substituting the values of the third term and the fifth term, we get:

a_3 = a * r^(3-1) = 3

a_5 = a * r^(5-1) = 27

Dividing the equation for the fifth term by the equation for the third term, we can eliminate the first term (a):

(a * r^(5-1)) / (a * r^(3-1)) = 27 / 3

Simplifying the equation, we get:

r^2 = 9

Taking the square root of both sides, we find:

r = ±3

Since the common ratio cannot be negative in this case (as it is a geometric progression), we take the positive value:

r = 3

Now that we have the common ratio, we can substitute it into the formula for the sum of the first five terms:

S_5 = a * (r^5 - 1) / (r - 1)

Substituting the values of the common ratio and the fifth term, we get:

27 = a * (3^5 - 1) / (3 - 1)

Simplifying the equation, we find:

27 = a * (243 - 1) / 2

27 = a * 242 / 2

Multiplying both sides by 2 and dividing by 242, we get:

a = 27 * 2 / 242

Simplifying the equation, we find:

a = 27 / 121

Now that we have the first term (a) and the common ratio (r), we can substitute them into the formula for the sum of the first five terms:

S_5 = a * (r^5 - 1) / (r - 1)

Substituting the values, we get:

S_5 = (27 / 121) * (3^5 - 1) / (3 - 1)

Simplifying the equation, we find:

S_5 = (27 / 121) * (243 - 1) / 2

S_5 = (27 / 121) * 242 / 2

S_5 = 27 * 242 / (121 * 2)

S_5 = 13.5

Therefore, the sum of the first five terms of the given geometric progression is 13.5.

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