Geometric Sequence

Front

What is a geometric sequence?

Back

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Front

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

Back

The nth term (a_n) of a geometric sequence can be calculated using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number.

Front

If the first term of a geometric sequence is 2 and the common ratio is 3, what is the 4th term?

Back

The 4th term is 2 * 3^(4-1) = 2 * 27 = 54.

Front

How do you find the sum of the first n terms of a geometric sequence?

Back

The sum (S_n) of the first n terms of a geometric sequence can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), where r is not equal to 1.

Front

What is the common ratio in a geometric sequence?

Back

The common ratio is the factor by which each term of the sequence is multiplied to get the next term.

Front

If a geometric sequence has a first term of 5 and a common ratio of 2, what is the 3rd term?

Back

The 3rd term is 5 * 2^(3-1) = 5 * 4 = 20.

Front

What happens to the terms of a geometric sequence if the common ratio is between 0 and 1?

Back

If the common ratio is between 0 and 1, the terms of the geometric sequence will decrease and approach zero.

Front

Calculate the amount after 3 years for an investment of $1,000 growing at 5% per year using a geometric sequence.

Back

The amount after 3 years is $1,000 * (1 + 0.05)^3 = $1,157.63.

Front

If a tree grows by 20% each year, what will be its height after 3 years if it starts at 2 meters?

Back

The height after 3 years is 2 * (1 + 0.20)^3 = 2.88 meters.

Front

What is the effect of a common ratio greater than 1 on a geometric sequence?

Back

If the common ratio is greater than 1, the terms of the geometric sequence will increase exponentially.