Gr12 Mathematics: Ch 1.4 Finite Geometric Series

What is the formula for the nth term of a geometric sequence?

$$T_n = a \cdot r^n$$

$$T_n = a \cdot r^{n-1}$$ (correct)

$$T_n = a \cdot r^{n+1}$$

$$T_n = a \cdot r^{n-2}$$

What is the purpose of multiplying both sides of the equation by r when deriving the formula for a finite geometric series?

To find the common ratio

To create a new equation to subtract from the original (correct)

To simplify the equation

To eliminate the first term

What is the condition for using the alternative formula for a finite geometric series?

r < 1

r = 1

r > 1 (correct)

r 1

What is the result of subtracting the second equation from the first when deriving the formula for a finite geometric series?

S_n - rS_n = a - ar^n Signup and view all the answers

What is the general formula for the sum of a finite geometric series?

$$S_n = rac{a(1 - r^n)}{1 - r}$$ Signup and view all the answers

What is the term used to describe the constant value by which each term in a geometric sequence is multiplied to obtain the next term?

Common ratio Signup and view all the answers

What is the value of the common ratio 'r' in a geometric sequence if the second term is 6 and the first term is 3?

$r = 2$ Signup and view all the answers

What is the sum of the first 5 terms of a geometric sequence if the first term is 2 and the common ratio is 3?

$242$ Signup and view all the answers

Which of the following is NOT a condition for using the alternative formula for a finite geometric series?

$r < 1$ Signup and view all the answers

What is the value of 'a' in a geometric sequence if the sum of the first 4 terms is 15 and the common ratio is 2?

$15/7$ Signup and view all the answers

What is the formula for the sum of a finite geometric series when the common ratio 'r' is less than 1?

$a(1 - r^n) / (1 - r)$ Signup and view all the answers

What is the value of the sum of the first 3 terms of a geometric sequence if the first term is 2 and the common ratio is -2?

$-14$ Signup and view all the answers

If the sum of the first n terms of a geometric sequence is 15 and the common ratio is 2, what can be said about the value of the first term?

The first term cannot be determined. Signup and view all the answers

What happens to the value of S_n as n approaches infinity if |r| > 1?

S_n grows without bound. Signup and view all the answers

If the sum of the first 5 terms of a geometric sequence is 242 and the common ratio is 3, what is the value of the first term?

2/3 Signup and view all the answers

What is the relationship between the sum of the first n terms of a geometric sequence and the sum of the first (n - 1) terms?

The sum of the first n terms is always equal to the sum of the first (n - 1) terms multiplied by the common ratio. Signup and view all the answers

If the sum of the first n terms of a geometric sequence is 15 and the sum of the first (n - 1) terms is 7.5, what is the value of the common ratio?

3 Signup and view all the answers

What is the sum of the first n terms of a geometric sequence if the common ratio is -1 and the first term is 1?

1 for all n Signup and view all the answers