Sum and Nth Term of Geometric Sequence Calculator

Welcome to use the sum and Nth term of geometric sequence calculator, this calculator solves two problems: one is to calculate the sum of geometric sequence; the other is to calculate the Nth term of the geometric sequence.

A geometric sequence refers to a sequence in which the ratio of each item to its previous item is equal to the same constant starting from the second item, commonly expressed as GP. This constant is called the common ratio and is usually denoted by the letter q.

For example, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 is a geometric sequence with a common ratio of 2.

When q = 1, the geometric sequence is a constant sequence. Such as 5, 5, 5, 5 is a constant sequence, the common ratio is 1.

Like the sum of the arithmetic sequence, there are two ways to calculate the sum of the geometric sequence.

Method 1: know the first item, the last item and the common ratio of the geometric sequence. The formula is as follows:

S_n = a₁ – a_n * q1 – q

Among them, a₁ represents the first term of the geometric sequence, a_n represents the last term of the geometric sequence, and q is the common ratio.

For example, 1, 3, 9, 27, 81, 243 is a geometric sequence with a common ratio of 3. The first term is 1, the last term is 243. So, use the formula to calculate their sum is:

S_n = a₁ – a_n * q1 – q = 1 – 243 * 3 1 – 3 = 364

Method 2: when the last term is unknown. It is calculated by using the first term, common ratio and number of terms. the formula is:

S_n = a₁ * (1 – qⁿ)1 – q

Among them, a₁ represents the first term of the geometric sequence, n represents the length of the geometric sequence, and q is the common ratio.

For example, a geometric sequence with 10 terms, the first term is 2 and the common ratio is 4. What is the sum of this geometric sequence?

S_n = a₁ * (1 – qⁿ)1 – q = 2 * (1 – 4¹⁰)1 – 4 = 699050

Of course, there is another situation that needs special attention. When the common ratio is 1, the geometric sequence is a constant sequence, so their sum is the first item multiplied by N. The formula is as follows:

S_n = n * a₁

a₁ represents the first term of the geometric sequence and n represents the length of the geometric sequence.

The constant sequence as mentioned above, 5, 5, 5, 5. The sum of them is 4 * 5 = 20.

If the common ratio of the geometric sequence is 1, then the Nth term is the same as the first term.

a_n = a₁

When the common ratio is not 1, the Nth term of the geometric sequence is:

a_n = a₁ * q^n-1

a₁ represents the first term of the geometric sequence, n represents the number of the Nth term, and q is the common ratio.

As the example mentioned in method 2 above: A geometric sequence with 10 terms, the first term is 2 and the common ratio is 4. What is the 10th term of this geometric sequence?

a_n = a₁ * q^n-1 = 2 * 4^10-1 = 524288

Whether calculating the sum of geometric sequence or calculating the Nth term of the geometric sequence, you must keep the formula in mind. If you are not familiar with formulas, or do not know how to use formulas, it will be a big trouble. Of course, the power calculation is also difficult. So, we developed this free online tool – Sum and Nth Term of Geometric Sequence Calculator, which can help you calculate the answer in a few milliseconds. Next, let’s take a look at how this tool is used.

The procedure to use the sum and Nth term of geometric sequence calculator is as follows: