What is the formula which shows how you calculate the nth term?

The formula to calculate the nth term depends on the type of sequence you're dealing with. The most common type is an arithmetic sequence.

Arithmetic Sequences

For an arithmetic sequence, the nth term (a_n) can be calculated using the following formula:

a_n = a₁ + (n - 1)d

Where:

• a_n is the nth term you want to find.
• a₁ is the first term of the sequence.
• n is the position of the term in the sequence (e.g., 5th term, 10th term).
• d is the common difference between consecutive terms.

Example:

Consider the arithmetic sequence: 2, 5, 8, 11, ...

Here, a₁ = 2 and d = 3 (since 5 - 2 = 3, 8 - 5 = 3, and so on).

To find the 10th term (a₁₀), we use the formula:

a₁₀ = 2 + (10 - 1) 3
a₁₀ = 2 + (9) 3
a₁₀ = 2 + 27
a₁₀ = 29

Therefore, the 10th term of the sequence is 29.

Geometric Sequences

For a geometric sequence, the nth term (a_n) can be calculated using the following formula:

*a_n = a₁ r^(n-1)**

Where:

• a_n is the nth term you want to find.
• a₁ is the first term of the sequence.
• n is the position of the term in the sequence.
• r is the common ratio between consecutive terms.

Example:

Consider the geometric sequence: 3, 6, 12, 24, ...

Here, a₁ = 3 and r = 2 (since 6 / 3 = 2, 12 / 6 = 2, and so on).

To find the 7th term (a₇), we use the formula:

a₇ = 3 2^(7-1)
a₇ = 3 2⁶
a₇ = 3 * 64
a₇ = 192

Therefore, the 7th term of the sequence is 192.

Conclusion

The formula to find the nth term depends on whether the sequence is arithmetic or geometric. For arithmetic sequences, use a_n = a₁ + (n - 1)d. For geometric sequences, use a_n = a₁ * r^(n-1).