$\begin{array}{l}
-21,-14,-7,0,7, \ldots \
a_{18}=[?] \
\text { Hint: } a_n=a_1+(n-1) d
\end{array}$
Answer (2)
Identify the first term a 1 = − 21 and the common difference d = 7 .
Use the formula for the nth term of an arithmetic sequence: a n = a 1 + ( n − 1 ) d .
Substitute n = 18 , a 1 = − 21 , and d = 7 into the formula: a 18 = − 21 + ( 18 − 1 ) × 7 .
Calculate a 18 = − 21 + 17 × 7 = − 21 + 119 = 98 , so the final answer is 98 .
Explanation
Understanding the Problem We are given an arithmetic sequence and asked to find the 18th term. The sequence is: − 21 , − 14 , − 7 , 0 , 7 , … We are also given the formula for the nth term of an arithmetic sequence: a n = a 1 + ( n − 1 ) d where a 1 is the first term, n is the term number, and d is the common difference.
Finding the First Term and Common Difference First, we need to identify the first term, a 1 , and the common difference, d , of the sequence. From the given sequence, we can see that the first term is a 1 = − 21 . To find the common difference, we can subtract any term from its subsequent term. For example, − 14 − ( − 21 ) = − 14 + 21 = 7 − 7 − ( − 14 ) = − 7 + 14 = 7 0 − ( − 7 ) = 0 + 7 = 7 7 − 0 = 7 So, the common difference is d = 7 .
Applying the Formula Now we can use the formula to find the 18th term, a 18 . We have a 1 = − 21 , d = 7 , and n = 18 . Plugging these values into the formula, we get: a 18 = a 1 + ( n − 1 ) d = − 21 + ( 18 − 1 ) × 7 = − 21 + ( 17 ) × 7
Calculating the 18th Term Now, we calculate the value of a 18 : a 18 = − 21 + ( 17 × 7 ) = − 21 + 119 = 98 Therefore, the 18th term of the sequence is 98.
Final Answer The 18th term of the arithmetic sequence is 98.
Examples
Arithmetic sequences are useful in many real-life situations. For example, if you save a fixed amount of money each month, the total amount you have saved over time forms an arithmetic sequence. Understanding arithmetic sequences can help you predict how much money you will have saved after a certain number of months. Another example is calculating the number of seats in a stadium where each row has a fixed number of additional seats compared to the previous row. This knowledge is also applicable in calculating simple interest, where the interest earned each year is constant.
To find the 18th term of the arithmetic sequence starting with -21 and having a common difference of 7, we use the formula a n = a 1 + ( n − 1 ) d . By substituting the values, we determine that a 18 = 98 . Thus, the 18th term is 98.
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