Which expression represents the sum of the first 30 terms of the sequence?
[tex]$\frac{-20+(30) 4}{2}$[/tex] [tex]$\frac{30(-20+4)}{2}$[/tex]
[tex]$\frac{4(-20+96)}{2}$[/tex] [tex]$\frac{30(-20+96)}{2}$[/tex]
Answer (2)
Recall the formula for the sum of an arithmetic series: S n = 2 n ( a 1 + a n ) .
Calculate the 30th term: a 30 = − 20 + ( 30 − 1 ) 4 = 96 .
Substitute the values into the sum formula: S 30 = 2 30 ( − 20 + 96 ) .
The expression representing the sum is: 2 30 ( − 20 + 96 ) .
Explanation
Understanding the Problem We are asked to find the expression that represents the sum of the first 30 terms of an arithmetic sequence. Let's break down the problem and identify the key components.
Recalling the Sum Formula The formula for the sum of the first n terms of an arithmetic sequence is given by: S n = 2 n ( a 1 + a n ) where:
S n is the sum of the first n terms,
n is the number of terms,
a 1 is the first term,
a n is the n th term.
Finding the 30th Term We are given that the first term a 1 = − 20 and the number of terms n = 30 . We also need to find the 30th term, a 30 . The common difference is d = 4 . The formula for the n th term of an arithmetic sequence is: a n = a 1 + ( n − 1 ) d
Calculating the 30th Term Substituting the given values into the formula for a 30 , we get: a 30 = − 20 + ( 30 − 1 ) 4 = − 20 + ( 29 ) 4 = − 20 + 116 = 96
Calculating the Sum of the First 30 Terms Now we can find the sum of the first 30 terms using the sum formula: S 30 = 2 30 ( a 1 + a 30 ) = 2 30 ( − 20 + 96 ) So, the expression for the sum of the first 30 terms is: S 30 = 2 30 ( − 20 + 96 )
Identifying the Correct Expression Comparing this expression with the given options, we find that the correct expression is 2 30 ( − 20 + 96 ) .
Examples
Understanding arithmetic sequences and their sums is useful in many real-life situations. For example, if you are saving money each month and increasing the amount you save by a fixed amount, you can use the sum of an arithmetic sequence to calculate your total savings after a certain number of months. Similarly, if a theater has rows of seats that increase by a fixed number of seats per row, you can calculate the total number of seats in the theater using the sum of an arithmetic sequence. This concept is also applicable in calculating loan repayments or any scenario involving a series of regular payments or increments.
The expression that represents the sum of the first 30 terms of the arithmetic sequence is 2 30 ( − 20 + 96 ) . This corresponds to the sum formula for arithmetic sequences after calculating the 30th term as 96. Hence, the correct choice is the expression 2 30 ( − 20 + 96 ) .
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