I have a few problems I need some help with.
The first is Evaluate the sum.
1. There is an E like symbol on the top is 10 on the bottom n=1 and to the right (7-2n)
2. The next question is find a10 for an arithmetic sequence with a1=6 and d=10
3. Find the number of terms in the series and then find the sum. Evaluate the sum.
16+23+30+37+44+.....+93
1. What you have is an arithmetic sequence where a=3 and d=-2
So, S10, the sum of the first 10 terms, is
S10 = 10/2 (2*3 + 9(-2)) = -60
same way for #2
3. looking at the terms, the difference is 7. Dividing the range of terms, we see that (93-16)/7 = 11
So, 7 is added to 16 11 times to get to 93. So, there are 12 terms.
Now find the sum the same way, using a=16,d=7,n=12
Sure! I'd be happy to help you with these problems. Let's break them down one by one.
Problem 1: Evaluating the sum
The sum you provided is represented using a summation notation (∑). The goal is to find the value of the sum by plugging in the numbers. Let's break down the summation notation step by step:
1. The symbol "∑" represents the sum.
2. The variable "n" represents the indices or terms over which the sum is taken. In this case, the range is from 1 to 10 (i.e., n=1 to n=10).
3. The expression "(7-2n)" represents the terms of the sum. For each term, we substitute the value of the index "n" into the expression.
To find the sum, we simply replace "n" with each value from 1 to 10, calculate the expression (7-2n), and add up all the results.
Here's how to calculate it step by step:
1. Start with n = 1:
Evaluate the expression (7-2n) = (7-2*1) = 5.
Add this term to the sum.
2. Move to n = 2:
Evaluate the expression (7-2n) = (7-2*2) = 3.
Add this term to the sum.
Repeat these steps for n = 3 to n = 10, calculating the expression (7-2n) for each value of n and adding the terms to the sum.
Problem 2: Finding a10 in an arithmetic sequence
In an arithmetic sequence, the value of each term is obtained from the previous term by adding a constant difference (d).
Given:
- a1 = 6 (which represents the first term)
- d = 10 (which represents the common difference)
To find a10, we need to use the formula for the nth term in an arithmetic sequence:
an = a1 + (n - 1) * d
In this case, n = 10.
Substituting the given values, we have:
a10 = a1 + (10 - 1) * d
= 6 + 9 * 10
= 6 + 90
= 96
Therefore, a10 = 96.
Problem 3: Finding the number of terms and evaluating the sum
In this problem, we have a series of numbers given, and we need to find the number of terms in the series as well as evaluate the sum.
The given series is: 16, 23, 30, 37, 44, ..., 93.
To find the number of terms:
1. We observe that starting from 16 and adding 7 to each subsequent term, we reach 93.
2. We need to find the number of terms between the first term (16) and the last term (93) with a common difference of 7.
Using the formula for the nth term of an arithmetic sequence:
an = a1 + (n - 1) * d
To find n (the number of terms), we rearrange the formula:
n = (an - a1) / d + 1
Substituting the given values:
n = (93 - 16) / 7 + 1
= 77 / 7 + 1
= 11 + 1
= 12
Hence, there are 12 terms in the series.
To evaluate the sum:
1. We have an arithmetic sequence with a common difference of 7.
2. We have 12 terms in the series, ranging from 16 to 93.
3. We can use the formula for the sum of an arithmetic series:
S = (n / 2) * (a1 + an)
Substituting the given values:
S = (12 / 2) * (16 + 93)
= 6 * 109
= 654
Therefore, the sum of the series is 654.
I hope this explanation helps you understand how to solve these problems!