Find the number of terms in a sequence; -15,-13,-11,-9...,13
Step 1: Determine the first term (a) and common difference (d).
In this sequence, the first term (a) is -15. The common difference (d) is 2 because each term is increasing by 2.
Step 2: Determine the last term (l).
The last term can be found using the formula:
l = a + (n - 1)d
where n is the number of terms.
Substitute the values we know:
13 = -15 + (n - 1)2
Simplify and solve for n:
28 = 2n - 2
30 = 2n
n = 15
Therefore, the number of terms in the sequence is 15.
Find the summation of 16 when r=1(5r-1).
To find the summation of 16 when r=1(5r-1), we need to plug in the values of r and add up the resulting terms:
r = 1: 5(1) - 1 = 4
16(4) = 64
So when r = 1, the term is 64.
The problem doesn't specify what values of r to sum over, so we'll assume it means to sum over all positive integers less than infinity (in other words, an infinite series). We can write this as:
Σ[16(5r-1)] from r=1 to infinity
To determine whether this series converges or diverges, we can use the ratio test:
a_n = 16(5n-1)
a_{n+1} = 16(5(n+1)-1) = 16(5n+4) = 80n + 64
|a_{n+1}/a_n| = |(80n + 64) / (16(5n-1))| = |(5n + 4)/(n - 1)|
As n approaches infinity, this ratio approaches 5. Since the ratio is greater than 1, the series diverges.
Therefore, the summation of 16 when r=1(5r-1) does not converge.
Evaluate Σr=1^16(r-1)
We can expand the sum as follows:
Σr=1^16(r-1) = (1-1) + (2-1) + (3-1) + ... + (15-1) + (16-1)
= 0 + 1 + 2 + ... + 14 + 15
This is an arithmetic series with a common difference of 1. We can use the formula for the sum of an arithmetic series to find the answer:
S = (n/2)(a + l)
where S is the sum, n is the number of terms, a is the first term, and l is the last term.
In this case, n = 16, a = 0, and l = 15. Plugging in the values, we get:
S = (16/2)(0 + 15) = 8(15) = 120
Therefore, Σr=1^16(r-1) = 120.
To find the number of terms in a sequence, you can use the formula for the nth term of an arithmetic sequence. The formula is given by:
nth term = first term + (n-1) * common difference
In this case, the first term is -15 and the common difference is 2 (adding 2 to each term). We need to find the value of 'n' for which the nth term is 13.
Let's substitute the values into the formula and solve for 'n':
13 = -15 + (n - 1) * 2
Moving -15 to the right side:
13 + 15 = (n - 1) * 2
28 = 2(n - 1)
Divide both sides by 2:
28 / 2 = n - 1
14 = n - 1
Adding 1 to both sides:
14 + 1 = n
n = 15
Therefore, there are 15 terms in the sequence -15, -13, -11, -9, ..., 13.
To find the number of terms in a sequence, we need to determine the pattern and use it to calculate the difference between the first and last term. In this sequence, the difference between consecutive terms is 2.
Let's break it down step by step:
1. Find the difference between the first and last term:
The first term is -15 and the last term is 13. Subtracting these values, we get 13 - (-15) = 13 + 15 = 28.
2. Find the common difference between terms:
As mentioned earlier, the common difference is 2.
3. Use the formula for the number of terms in an arithmetic sequence:
The formula to find the number of terms in an arithmetic sequence is:
Number of terms = (last term - first term) / common difference + 1
Applying the formula to our sequence, we have:
Number of terms = (13 - (-15)) / 2 + 1
= (13 + 15) / 2 + 1
= 28 / 2 + 1
= 14 + 1
= 15
Therefore, the number of terms in the given sequence -15, -13, -11, -9, ..., 13 is 15.