Question

How many terms in the GP 4, 3.6, 3.24, ... are needed so that the sum exceeds 35?​

Answer 1

At least 3 terms are needed for the sum of the given geometric progression to exceed 35.

Step-by-step explanation:

To find the number of terms needed for the sum of the geometric progression (GP) to exceed 35, we can use the formula for the sum of a GP:

Sn = a(1 - rn) / (1 - r)

Given that a = 4 and r = 0.9 (since each term is 90% of the previous term), we can set up the following equation:

4(1 - 0.9n) / (1 - 0.9) > 35

Simplifying this equation, we get:

0.1n < 0.275

To solve for n, we can take the logarithm of both sides:

n > log0.1(0.275)

Using a calculator, we find that log0.1(0.275) is approximately 2.0589. Therefore, we need at least 3 terms for the sum of the GP to exceed 35.

Answer 2

The minimum number of terms in the geometric progression needed to exceed 35 is 6.

Step-by-step explanation:

To find the number of terms in the geometric progression (GP) that exceeds 35, we need to determine the general formula for the terms of the GP. In a geometric progression, each term is obtained by multiplying the previous term by a constant ratio. Given the first term (a = 4) and the common ratio (r = 0.9), we can write the general formula for the terms of the GP as:

an = a * r(n-1)

Now, let's substitute the value of a and r into the formula and solve for n:

4 * 0.9(n-1) > 35

0.9(n-1) > 35/4

0.9(n-1) > 8.75

To solve this inequality, we can take the logarithm of both sides with base 0.9:

(n-1) > log0.9(8.75)

(n-1) > 4.018

n > 4.018 + 1

n > 5.018

Since we need a whole number of terms, the minimum number of terms required to exceed 35 is 6.