Express the series 11 + 18 + 27 +...+ 171 using sigma notation.
Usually you would take 27-18 and that's the step between numbers, right? (I think its the k-value?) But 27-18=9 and 18-11=7. Are we stepping by k+2 here? Like k, k+2 k+4, k+6...?
Please help me. I'm not confident in sigma notation, and this question is difficult for me. Reiny, Damon, and Bobpursley have all given me good feedback before, but any tutor is welcome to respond to this!
To express the series 11 + 18 + 27 + ... + 171 using sigma notation, we need to determine the pattern or formula that represents the terms.
Let's analyze the given series: 11, 18, 27, ... , 171.
We can observe that each term in the series is formed by multiplying consecutive odd numbers starting from 3.
The first term (11) is obtained by multiplying 3 and 4. The second term (18) is formed by multiplying 5 and 6, and so on.
In general, the nth term can be written as (2n + 1) × (2n + 2).
Now, let's rewrite the series using the formula:
11 + 18 + 27 + ... + 171
= (2 × 1 + 1) × (2 × 1 + 2) + (2 × 2 + 1) × (2 × 2 + 2) + ... + (2 × n + 1) × (2 × n + 2)
= Σ[(2n + 1) × (2n + 2)] for n = 1 to n = ?
The sigma notation represents the sum of the terms in the series. The lower limit (n = 1) indicates that we start from the first term, and the upper limit (?) indicates that we continue until a certain term (which we need to determine).
To determine the upper limit, we need to find the value of n when the nth term reaches 171.
(2n + 1) × (2n + 2) = 171
Expanding and rearranging, we get:
4n^2 + 6n + 2 = 171
4n^2 + 6n - 169 = 0
This is a quadratic equation. Solving it, we find that n ≈ 6.7.
Since n needs to be a whole number, we can take the upper limit as n = 6.
Therefore, the series can be expressed using sigma notation as:
Σ[(2n + 1) × (2n + 2)] for n = 1 to n = 6
To express the series 11 + 18 + 27 +...+ 171 using sigma notation, it helps to find the pattern in the series.
First, let's try to figure out the rule for generating the terms of this series. Looking at the series, we can see that each term is obtained by adding a constant difference to the previous term.
Starting with the first term, we have 11, and adding the constant difference of 7, we get the second term, 11 + 7 = 18. Adding the difference to the second term, we get the third term, 18 + 7 = 25. Carrying on this pattern, we can see that the rule for generating the terms is to add 7 to the previous term.
Now, let's determine the number of terms in this series. We want to find how many times we need to add 7 to reach the last term, which is 171. So we can set up an equation:
11 + (n - 1) * 7 = 171
Simplifying this equation, we have:
11 + 7n - 7 = 171
7n + 4 = 171
7n = 167
n = 167/7 = 23.857...
Since the number of terms must be a whole number, we should take the floor of n, which is 23.
So, the series has 23 terms.
Now, let's express this series using sigma notation:
The formula for the sum of an arithmetic series can be represented using sigma (Σ) notation as follows:
Σ(k=1 to n) [a + (k-1)d]
where:
- Σ denotes the sum
- k is the index variable that represents the term number
- a is the first term of the series
- d is the common difference between terms
- n is the number of terms in the series
In our case, a = 11, d = 7, and n = 23.
So the series 11 + 18 + 27 + ... + 171 can be expressed as:
Σ(k=1 to 23) [11 + (k-1)7]
Note that in this case, the index variable k does not represent the value of the term itself, but rather the position or number of the term.
I hope this explanation helps you understand how to express the given series using sigma notation.