Find the third term and the sum of the first 30 terms of the geometric progression 2, 3, ...

Alright, let's dive into this geometric progression! The first term is 2, and to find the third term, we can use the formula:

Term(n) = FirstTerm * CommonRatio^(n-1)

So, for the third term (n=3), we have:

Term(3) = 2 * (3/2)^(3-1)
= 2 * (3/2)^2
= 2 * 2.25
= 4.5

Now, to find the sum of the first 30 terms, we can use the formula for the sum of a geometric progression:

Sum(n) = FirstTerm * (1 - CommonRatio^n) / (1 - CommonRatio)

Plugging in the values, we get:

Sum(30) = 2 * (1 - (3/2)^30) / (1 - (3/2))
= 2 * (1 - 1126.99816895) / (-1/2)
= 2 * (-1126.99816895) / (-1/2)
= 2 * (-1126.99816895) * (-2)
= 4515.9926758

So, the third term is 4.5, and the sum of the first 30 terms is approximately 4515.99. Keep in mind that numbers have a funny way of multiplying and dividing when clowns get involved!

To find the third term of the geometric progression, we can use the formula for the nth term of a geometric progression:

an = a1 * r^(n-1),

where "an" represents the nth term, "a1" is the first term, "r" is the common ratio, and "n" is the term number.

In this case, the first term (a1) is 2 and the common ratio (r) can be found by dividing the second term by the first term:

r = 3 / 2 = 1.5.

Now we can find the third term (a3) using the formula:

a3 = a1 * r^(3-1) = 2 * 1.5^(2) = 2 * 2.25 = 4.5.

Therefore, the third term is 4.5.

To find the sum of the first 30 terms of a geometric progression, we can use the formula for the sum of an n-term geometric series:

Sn = a1 * (1 - r^n) / (1 - r),

where "Sn" represents the sum of the first n terms.

In this case, we need to find the sum of the first 30 terms, so:

a1 = 2,
r = 1.5,
n = 30.

Now we can substitute these values into the formula:

S30 = 2 * (1 - 1.5^30) / (1 - 1.5).

Using a calculator, we can calculate the sum:

S30 ≈ 49.9999999976.

Therefore, the sum of the first 30 terms is approximately 50.

To find the third term of a geometric progression, we need to determine the common ratio (r) first. In a geometric progression, each term is obtained by multiplying the preceding term by the common ratio.

In this case, we can find the common ratio by dividing the second term by the first term:

r = 3/2 = 1.5

Now, to find the third term, we can multiply the second term (3) by the common ratio:

Third term = 3 * 1.5 = 4.5

To find the sum of the first 30 terms, we can use the formula for the sum of a geometric progression:

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

where:
- a is the first term in the series
- r is the common ratio
- n is the number of terms we want to sum

Plugging in the values, we have:

a = 2 (first term)
r = 1.5 (common ratio)
n = 30 (number of terms)

S = 2 * (1 - 1.5^30) / (1 - 1.5)

After evaluating this expression, we can find the sum of the first 30 terms.

clearly, r = 3/2, so the next term is 9/2

And, as always, the sum of the first n terms is

Sn = a (r^n-1)/(r-1)
So plug in your numbers.