Find the 24th term of this arithmetic sequence.
[tex]\begin{array}{c}
-21,-14,-7,0,7, \ldots \
a_{24}=[?]\end{array}[/tex]
Hint: [tex]a_n=a_1+(n-1) d[/tex]
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 = 24 , a 1 = − 21 , and d = 7 into the formula: a 24 = − 21 + ( 24 − 1 ) ( 7 ) .
Calculate a 24 = − 21 + ( 23 ) ( 7 ) = − 21 + 161 = 140 , so the 24th term is 140 .
Explanation
Understanding the Problem We are given an arithmetic sequence and asked to find the 24th 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 Common Difference First, we need to find the common difference, d . We can find this by subtracting any term from its subsequent term. Let's subtract the first term from the second term: d = a 2 − a 1 = − 14 − ( − 21 ) = − 14 + 21 = 7 So, the common difference d = 7 .
Calculating the 24th Term Now we can use the formula to find the 24th term, a 24 . We know that a 1 = − 21 , n = 24 , and d = 7 . Plugging these values into the formula, we get: a 24 = a 1 + ( n − 1 ) d = − 21 + ( 24 − 1 ) ( 7 ) = − 21 + ( 23 ) ( 7 ) = − 21 + 161 = 140 Therefore, the 24th term of the arithmetic sequence is 140.
Final Answer The 24th term of the arithmetic sequence is 140.
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 how to find a specific term in a sequence can help you predict how much money you will have saved after a certain number of months. Another example is calculating the height of stacked objects where each object adds a constant height to the stack.
The 24th term of the arithmetic sequence is 140. This is calculated using the formula for the nth term of an arithmetic sequence, substituting the first term and the common difference. After performing the necessary calculations, we find that a_{24} equals 140.
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