Understanding the 10th Term in a Sequence: Calculating an n2 - 1

When working with sequences and series in mathematics, one of the tasks you might be asked to perform is finding the value of a specific term. This article will clarify how to calculate the 10th term of a sequence given by the formula an n2 - 1.

Formula and Being Cautious of Order of Operations

The given formula for the nth term of the sequence is an n2 - 1. If we are to find the 10th term, we need to substitute n 10 into this formula. It is important to note that the order of operations is crucial in such arithmetic expressions.

Correct Method: a10 102 - 1

To correctly calculate the 10th term:

Substitute n 10 into the formula an n2 - 1. Calculate 102, which equals 100. Subtract 1 from 100, resulting in 99.

Therefore, the 10th term, a10, is 99.

Common Mistakes and Misinterpretations

It is not uncommon to encounter mistakes or misinterpretations when working with formulas involving fractions and exponents. Let's examine some common issues that could lead to incorrect results:

Mistaken Formula: a10 102 - 1/102

A potential mistake is interpreting the formula as a10 102 - 1/102. This would lead to:

Calculate 102, which equals 100. Calculate 1/102, which equals 0.01. Subtract 0.01 from 100, resulting in 99.99.

Clearly, this is not the correct interpretation, as we need to subtract 1, not 1/100.

Misinterpreted Formula: a10 (102 - 1) / 102

Another possible mistake arises if we interpret the formula as a10 (102 - 1) / 102. This would lead to:

Calculate 102, which equals 100. Calculate 100 - 1, which equals 99. Divide 99 by 100, resulting in 0.99.

This result is also incorrect as it is not the expected subtraction of 1 from the square of 10.

Conclusion

The correct way to find the 10th term of the sequence defined by an n2 - 1 is a10 102 - 1 which equals 99. This calculation is straightforward once you are aware of the order of operations and correctly interpret the given mathematical expression.

Understanding sequence formulas and correctly applying arithmetic operations is crucial for solving problems in algebra and beyond. If you encounter similar problems, always double-check your formula and the operations involved to ensure accurate results.