Find The First Term And The Common Difference Of An AP In Which The Sum Of The First 11terms Is 352and The Sum Of The Next 10 Terms Is 845.
=====================================================
Introduction
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. In this article, we will discuss how to find the first term and the common difference of an AP given the sum of the first 11 terms and the sum of the next 10 terms.
Formula for the Sum of n Terms of an AP
The formula for the sum of n terms of an AP is given by:
S_n = n/2 [2a + (n-1)d]
where S_n is the sum of n terms, a is the first term, n is the number of terms, and d is the common difference.
Given Information
We are given that the sum of the first 11 terms is 352 and the sum of the next 10 terms is 845.
Step 1: Find the Sum of the First 11 Terms
Let's use the formula for the sum of n terms of an AP to find the sum of the first 11 terms.
S_11 = 11/2 [2a + (11-1)d] S_11 = 11/2 [2a + 10d] S_11 = 5.5 [2a + 10d]
We are given that S_11 = 352. So, we can write:
5.5 [2a + 10d] = 352
Step 2: Find the Sum of the Next 10 Terms
The sum of the next 10 terms is given by:
S_21 = S_11 + S_10
where S_21 is the sum of the first 21 terms, S_11 is the sum of the first 11 terms, and S_10 is the sum of the next 10 terms.
We are given that S_21 = S_11 + 845. So, we can write:
S_21 = 352 + 845 S_21 = 1197
Step 3: Use the Formula for the Sum of n Terms of an AP to Find the Sum of the First 21 Terms
We can use the formula for the sum of n terms of an AP to find the sum of the first 21 terms.
S_21 = 21/2 [2a + (21-1)d] S_21 = 21/2 [2a + 20d] S_21 = 10.5 [2a + 20d]
We are given that S_21 = 1197. So, we can write:
10.5 [2a + 20d] = 1197
Step 4: Solve the Equations to Find the First Term and the Common Difference
We have two equations:
5.5 [2a + 10d] = 352 10.5 [2a + 20d] = 1197
We can simplify these equations by dividing both sides by 5.5 and 10.5 respectively.
2a + 10d = 64 2a + 20d = 112.36
Now, we can subtract the first equation from the second equation to eliminate the variable a.
10d = 48.36 d = 4.836
Step 5: Find the First Term
Now that we have the common difference, we can find the first term by substituting the value of d into one of the equations.
2a + 10d = 64 2a + 10(4.836) = 64 2a + 48.36 = 64 2a = 15.64 a = 7.82
Conclusion
In this article, we discussed how to find the first term and the common difference of an AP given the sum of the first 11 terms and the sum of the next 10 terms. We used the formula for the sum of n terms of an AP and solved the equations to find the first term and the common difference. The first term is 7.82 and the common difference is 4.836.
Example Use Case
This problem can be used in a variety of real-world scenarios, such as:
- Finding the first term and the common difference of a sequence of numbers that represent the population of a country over a period of time.
- Finding the first term and the common difference of a sequence of numbers that represent the prices of a product over a period of time.
- Finding the first term and the common difference of a sequence of numbers that represent the scores of a student in a class over a period of time.
Formula for the Sum of n Terms of an AP
The formula for the sum of n terms of an AP is given by:
S_n = n/2 [2a + (n-1)d]
where S_n is the sum of n terms, a is the first term, n is the number of terms, and d is the common difference.
Formula for the nth Term of an AP
The formula for the nth term of an AP is given by:
a_n = a + (n-1)d
where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.
Formula for the Sum of an Infinite AP
The formula for the sum of an infinite AP is given by:
S = a/(1-r)
where S is the sum, a is the first term, and r is the common ratio.
Formula for the Common Ratio of an AP
The formula for the common ratio of an AP is given by:
r = d/a
where r is the common ratio, d is the common difference, and a is the first term.
Formula for the nth Term of a GP
The formula for the nth term of a GP is given by:
a_n = a*r^(n-1)
where a_n is the nth term, a is the first term, n is the term number, and r is the common ratio.
Formula for the Sum of an Infinite GP
The formula for the sum of an infinite GP is given by:
S = a/(1-r)
where S is the sum, a is the first term, and r is the common ratio.
Formula for the Common Ratio of a GP
The formula for the common ratio of a GP is given by:
r = a_n/a_(n-1)
where r is the common ratio, a_n is the nth term, and a_(n-1) is the (n-1)th term.
Formula for the Sum of n Terms of a GP
The formula for the sum of n terms of a GP is given by:
S_n = a*(1-r^n)/(1-r)
where S_n is the sum of n terms, a is the first term, r is the common ratio, and n is the number of terms.
Formula for the nth Term of a HP
The formula for the nth term of a HP is given by:
a_n = a + (n-1)d
where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.
Formula for the Sum of n Terms of a HP
The formula for the sum of n terms of a HP is given by:
S_n = n/2 [2a + (n-1)d]
where S_n is the sum of n terms, a is the first term, n is the number of terms, and d is the common difference.
Formula for the Sum of an Infinite HP
The formula for the sum of an infinite HP is given by:
S = a/(1-r)
where S is the sum, a is the first term, and r is the common ratio.
Formula for the Common Ratio of a HP
The formula for the common ratio of a HP is given by:
r = d/a
where r is the common ratio, d is the common difference, and a is the first term.
Formula for the Sum of n Terms of a QP
The formula for the sum of n terms of a QP is given by:
S_n = n/2 [2a + (n-1)d]
where S_n is the sum of n terms, a is the first term, n is the number of terms, and d is the common difference.
Formula for the Sum of an Infinite QP
The formula for the sum of an infinite QP is given by:
S = a/(1-r)
where S is the sum, a is the first term, and r is the common ratio.
Formula for the Common Ratio of a QP
The formula for the common ratio of a QP is given by:
r = d/a
where r is the common ratio, d is the common difference, and a is the first term.
Formula for the Sum of n Terms of a SP
The formula for the sum of n terms of a SP is given by:
S_n = n/2 [2a + (n-1)d]
where S_n is the sum of n terms, a is the first term, n is the number of terms, and d is the common difference.
Formula for the Sum of an Infinite SP
The formula for the sum of an infinite SP is given by:
S = a/(1-r)
where S is the sum, a is the first term, and r is the common ratio.