Find The First Term When Given The Sum Of First 15 Terms As 450 And Common Difference Is 5
Introduction
Arithmetic sequences are a fundamental concept in mathematics, and they have numerous applications in various fields, including finance, engineering, and science. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. In this article, we will discuss how to find the first term of an arithmetic sequence when given the sum of the first 15 terms and the common difference.
Understanding Arithmetic Sequences
An arithmetic sequence can be represented by the formula:
a, a + d, a + 2d, a + 3d, ...
where 'a' is the first term, and 'd' is the common difference. For example, if the first term is 2 and the common difference is 3, the sequence would be:
2, 5, 8, 11, 14, ...
Finding the Sum of an Arithmetic Sequence
The sum of the first n terms of an arithmetic sequence can be calculated using the formula:
S_n = n/2 * (2a + (n-1)d)
where S_n is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
Given Information
We are given that the sum of the first 15 terms of an arithmetic sequence is 450, and the common difference is 5. We need to find the first term of the sequence.
Using the Formula to Find the First Term
We can use the formula for the sum of an arithmetic sequence to find the first term. Plugging in the given values, we get:
450 = 15/2 * (2a + (15-1)5)
Simplifying the equation, we get:
450 = 7.5 * (2a + 70)
Dividing both sides by 7.5, we get:
60 = 2a + 70
Subtracting 70 from both sides, we get:
-10 = 2a
Dividing both sides by 2, we get:
-5 = a
Therefore, the first term of the arithmetic sequence is -5.
Conclusion
In this article, we discussed how to find the first term of an arithmetic sequence when given the sum of the first 15 terms and the common difference. We used the formula for the sum of an arithmetic sequence to find the first term. The first term of the sequence is -5.
Example Problems
- Find the first term of an arithmetic sequence when the sum of the first 10 terms is 200 and the common difference is 2.
- Find the first term of an arithmetic sequence when the sum of the first 20 terms is 1000 and the common difference is 3.
Solutions
- Using the formula for the sum of an arithmetic sequence, we get:
200 = 10/2 * (2a + (10-1)2)
Simplifying the equation, we get:
200 = 5 * (2a + 18)
Dividing both sides by 5, we get:
40 = 2a + 18
Subtracting 18 from both sides, we get:
22 = 2a
Dividing both sides by 2, we get:
11 = a
Therefore, the first term of the arithmetic sequence is 11.
- Using the formula for the sum of an arithmetic sequence, we get:
1000 = 20/2 * (2a + (20-1)3)
Simplifying the equation, we get:
1000 = 10 * (2a + 57)
Dividing both sides by 10, we get:
100 = 2a + 57
Subtracting 57 from both sides, we get:
43 = 2a
Dividing both sides by 2, we get:
21.5 = a
Therefore, the first term of the arithmetic sequence is 21.5.
Final Thoughts
Arithmetic sequences are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we discussed how to find the first term of an arithmetic sequence when given the sum of the first 15 terms and the common difference. We used the formula for the sum of an arithmetic sequence to find the first term. The first term of the sequence is -5. We also provided example problems and solutions to help readers understand the concept better.
Introduction
Arithmetic sequences are a fundamental concept in mathematics, and they have numerous applications in various fields. In our previous article, we discussed how to find the first term of an arithmetic sequence when given the sum of the first 15 terms and the common difference. In this article, we will provide a Q&A section to help readers understand the concept better.
Q&A
Q1: What is an arithmetic sequence?
A1: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.
Q2: How do I find the sum of an arithmetic sequence?
A2: The sum of the first n terms of an arithmetic sequence can be calculated using the formula:
S_n = n/2 * (2a + (n-1)d)
where S_n is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
Q3: How do I find the first term of an arithmetic sequence when given the sum of the first 15 terms and the common difference?
A3: We can use the formula for the sum of an arithmetic sequence to find the first term. Plugging in the given values, we get:
S_n = n/2 * (2a + (n-1)d)
Simplifying the equation, we get:
450 = 15/2 * (2a + (15-1)5)
Dividing both sides by 15/2, we get:
60 = 2a + 70
Subtracting 70 from both sides, we get:
-10 = 2a
Dividing both sides by 2, we get:
-5 = a
Therefore, the first term of the arithmetic sequence is -5.
Q4: What is the formula for the nth term of an arithmetic sequence?
A4: The formula for the nth term of an arithmetic sequence is:
a_n = a + (n-1)d
where a_n is the nth term, a is the first term, d is the common difference, and n is the term number.
Q5: How do I find the common difference of an arithmetic sequence?
A5: The common difference of an arithmetic sequence can be found by subtracting any two consecutive terms. For example, if the sequence is 2, 5, 8, 11, ..., the common difference is 3.
Q6: What is the formula for the sum of the first n terms of an arithmetic sequence?
A6: The formula for the sum of the first n terms of an arithmetic sequence is:
S_n = n/2 * (2a + (n-1)d)
where S_n is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
Q7: How do I find the number of terms in an arithmetic sequence?
A7: The number of terms in an arithmetic sequence can be found by using the formula:
n = (S_n - a) / d
where n is the number of terms, S_n is the sum of the first n terms, a is the first term, and d is the common difference.
Q8: What is the formula for the average of an arithmetic sequence?
A8: The formula for the average of an arithmetic sequence is:
average = (a + a_n) / 2
where average is the average of the sequence, a is the first term, and a_n is the nth term.
Conclusion
Arithmetic sequences are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we provided a Q&A section to help readers understand the concept better. We covered topics such as finding the sum of an arithmetic sequence, finding the first term of an arithmetic sequence, and finding the common difference of an arithmetic sequence. We also provided formulas for the nth term, the sum of the first n terms, and the average of an arithmetic sequence.