Problem 56
Question
Find each sum. The sum of the first 46 terms of the sequence $$ 2,-1,-4,-7, \ldots $$
Step-by-Step Solution
Verified Answer
The sum of the first 46 terms is -3013.
1Step 1: Identify the Sequence Type
First, identify the type of sequence given. The sequence 2, -1, -4, -7, ... has a common difference of -3 (each term is 3 less than the previous term), which means it is an arithmetic sequence.
2Step 2: Use the Sum of an Arithmetic Sequence Formula
The formula to find the sum of the first n terms in an arithmetic sequence is \[ S_n = \frac{n}{2} (a + l) \] where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.
3Step 3: Determine the Last Term
To find the last term (\( l \)) in the sequence, use the formula for the n-th term of an arithmetic sequence, \[ a_n = a + (n-1) d \]. Here, \( a \) is the first term (2), \( n \) is 46, and \( d \) is the common difference (-3). Plug these values in: \[ a_{46} = 2 + (46-1) (-3) \] \[ a_{46} = 2 + 45 (-3) \] \[ a_{46} = 2 - 135 \] \[ a_{46} = -133 \]. So, the last term (\( l \)) is -133.
4Step 4: Calculate the Sum of the Sequence
Now, substitute \( n = 46 \), \( a = 2 \), and \( l = -133 \) into the sum formula: \[ S_{46} = \frac{46}{2} (2 + (-133)) \] \[ S_{46} = 23 (-131) \] \[ S_{46} = -3013 \].
Key Concepts
arithmetic progressionsequence formulassum calculationn-th term
arithmetic progression
An arithmetic progression (or arithmetic sequence) is a series of numbers in which each term after the first is found by adding a constant to the previous term. This constant is known as the common difference. For instance, in the sequence 2, -1, -4, -7, ..., the common difference is -3.
We calculate the common difference by subtracting any term from the term following it. To confirm it's an arithmetic progression, check if the difference remains consistent across the sequence.
Arithmetic progressions are crucial for recognizing patterns and calculating sums over series.
We calculate the common difference by subtracting any term from the term following it. To confirm it's an arithmetic progression, check if the difference remains consistent across the sequence.
Arithmetic progressions are crucial for recognizing patterns and calculating sums over series.
sequence formulas
When dealing with arithmetic sequences, there are essential formulas you need to know. For the sum of the first n terms, we use:
\[ S_n = \frac{n}{2} (a + l) \] Here, \(S_n\) is the sum of the first n terms, \(a\) is the first term, and \(l\) is the last term of the sequence.
Another critical formula is for the n-th term, given by:
\[ a_n = a + (n-1) d \] In this formula, \(a_n\) represents the n-th term, \(a\) refers to the first term, \(n\) gives the number of terms, and \(d\) denotes the common difference.
Using these formulas, we can effectively handle a wide range of arithmetic sequence problems and calculations.
\[ S_n = \frac{n}{2} (a + l) \] Here, \(S_n\) is the sum of the first n terms, \(a\) is the first term, and \(l\) is the last term of the sequence.
Another critical formula is for the n-th term, given by:
\[ a_n = a + (n-1) d \] In this formula, \(a_n\) represents the n-th term, \(a\) refers to the first term, \(n\) gives the number of terms, and \(d\) denotes the common difference.
Using these formulas, we can effectively handle a wide range of arithmetic sequence problems and calculations.
sum calculation
Calculating the sum of an arithmetic sequence involves multiple steps. Let's break it down:
\[ S_{46} = \frac{46}{2} (2 + (-133)) = -3013 \] Thus, extensive understanding of each term's value helps in accurate sum calculation.
- First, identify the type of sequence and confirm it's arithmetic by verifying a common difference.
- Next, find the number of terms \(n\), the first term \(a\), and the last term \(l\) of the sequence.
- Use the sum formula \[ S_n = \frac{n}{2} (a + l) \] to get the sum.
\[ S_{46} = \frac{46}{2} (2 + (-133)) = -3013 \] Thus, extensive understanding of each term's value helps in accurate sum calculation.
n-th term
Finding the n-th term in an arithmetic sequence is quite straightforward with the n-th term formula. Following the general form:
\[ a_n = a + (n-1) d \] allows you to plug in the known values to reach the specific term's value. It's pivotal for defining the sequence's progression pattern.
For instance, in the sequence 2, -1, -4, -7, ..., calculate the 46th term using:
\[ a_{46} = 2 + (46-1) (-3) = -133 \] Understanding this helps in solving for any term within the sequence, especially beneficial when addressing larger n values or ensuring accurate sum calculations.
\[ a_n = a + (n-1) d \] allows you to plug in the known values to reach the specific term's value. It's pivotal for defining the sequence's progression pattern.
For instance, in the sequence 2, -1, -4, -7, ..., calculate the 46th term using:
\[ a_{46} = 2 + (46-1) (-3) = -133 \] Understanding this helps in solving for any term within the sequence, especially beneficial when addressing larger n values or ensuring accurate sum calculations.
Other exercises in this chapter
Problem 56
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