Find the Next Term 70 , 65 , 60

Solution:

Step 1:

Identify the pattern in the sequence. The sequence is decreasing by 5 each time, indicating it's an arithmetic sequence. The common difference (d) is -5, which can be expressed as $a_{n} = a_{1} + (n - 1)d$ where $d = -5$.

Step 2:

Recall the general formula for the nth term of an arithmetic sequence: $a_{n} = a_{1} + (n - 1)d$.

Step 3:

Plug in the known values, with $a_{1} = 70$ and $d = -5$, into the formula: $a_{n} = 70 + (n - 1)(-5)$.

Step 4:

Perform algebraic simplification:

Step 4.1:

Distribute the common difference: $a_{n} = 70 - 5(n - 1)$.

Step 4.2:

Simplify the expression by multiplying: $a_{n} = 70 - 5n + 5$.

Step 5:

Combine like terms to simplify further: $a_{n} = 75 - 5n$.

Step 6:

Determine the fourth term ($a_{4}$) by substituting $n = 4$: $a_{4} = 75 - 5(4)$.

Step 7:

Calculate the multiplication: $a_{4} = 75 - 20$.

Step 8:

Finally, add the numbers to find the next term: $a_{4} = 55$.

Knowledge Notes:

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference, called the common difference (d), to the previous term. The nth term of an arithmetic sequence can be found using the formula:

$$a_{n} = a_{1} + (n - 1)d$$

where:

• $a_{n}$ is the nth term of the sequence,
• $a_{1}$ is the first term of the sequence,
• $d$ is the common difference,
• $n$ is the term number.

In this problem, the sequence is decreasing, so the common difference is negative. When dealing with arithmetic sequences, it is important to correctly identify the common difference and apply the formula accurately to find the desired term. Algebraic simplification is often required to find the exact term in the sequence.