The sum of an arithmetic sequence can be found by the formula [tex]$S_n = \frac{1}{2}n(2a + (n-1)d)$[/tex]. Use the formula to find the sum of the first 10 terms of the sequence: [tex]$\{-3, 2, 7, 12, 17, 22, \ldots\}$[/tex]
Answer (1)
Identify the first term a = − 3 , the common difference d = 5 , and the number of terms n = 10 .
Substitute these values into the formula for the sum of an arithmetic sequence: S n = 2 1 n ( 2 a + ( n − 1 ) d ) .
Calculate the sum: S 10 = 2 1 ( 10 ) ( 2 ( − 3 ) + ( 10 − 1 ) ( 5 )) = 5 ( − 6 + 45 ) = 5 ( 39 ) = 195 .
The sum of the first 10 terms of the sequence is 195 .
Explanation
Understanding the Problem We are given an arithmetic sequence { − 3 , 2 , 7 , 12 , 17 , 22 , … } and we want to find the sum of the first 10 terms using the formula S n = 2 1 n ( 2 a + ( n − 1 ) d ) , where S n is the sum of the first n terms, a is the first term, and d is the common difference.
Identifying the Values First, we need to identify the values of a , d , and n . The first term a is the first number in the sequence, which is − 3 . The common difference d is the difference between consecutive terms. We can find d by subtracting the first term from the second term: d = 2 − ( − 3 ) = 5 . The number of terms n is given as 10.
Substituting the Values Now, we substitute the values a = − 3 , d = 5 , and n = 10 into the formula: S 10 = 2 1 ( 10 ) ( 2 ( − 3 ) + ( 10 − 1 ) ( 5 ))
Calculating the Sum Next, we simplify the expression: S 10 = 2 1 ( 10 ) ( − 6 + ( 9 ) ( 5 )) S 10 = 5 ( − 6 + 45 ) S 10 = 5 ( 39 ) S 10 = 195
Final Answer Therefore, the sum of the first 10 terms of the arithmetic sequence is 195.
Examples
Arithmetic sequences and their sums are useful in various real-life scenarios. For example, consider a savings plan where you deposit a fixed amount of money each month. If the deposits increase by a constant amount each month, the total savings over a period can be calculated using the sum of an arithmetic sequence. This helps in financial planning and forecasting.
