The sum of the nth and 3rd term of an AP is 42. If the difference between the 6th and 3rd term is 12, find the common difference (d) and the 20th term.
Answer (2)
The common difference d is found using the difference between the 6th and 3rd terms: 3 d = 12 , so d = 4 .
The first term a is found using the sum of the first and 20th terms: 2 a + 19 d = 42 , substituting d = 4 gives a = − 17 .
The 20th term a 20 is calculated using a 20 = a + 19 d , substituting a = − 17 and d = 4 gives a 20 = 59 .
The common difference is 4 , the first term is − 17 , and the 20th term is 59 .
Explanation
Understanding the Problem We are given that the sum of the first term and the 20th term of an arithmetic progression (AP) is 42, and the difference between the 6th and 3rd terms is 12. Our goal is to find the common difference, the first term, and the 20th term of this AP.
Defining Variables and Formulas Let a be the first term and d be the common difference of the AP. The n th term of an AP is given by a n = a + ( n − 1 ) d .
Using the Sum of First and 20th Term The sum of the first term and the 20th term is 42, so we have: a + a 20 = 42 Substituting a 20 = a + 19 d , we get: a + ( a + 19 d ) = 42 2 a + 19 d = 42
Using the Difference Between 6th and 3rd Term The difference between the 6th and 3rd terms is 12, so we have: a 6 − a 3 = 12 Substituting a 6 = a + 5 d and a 3 = a + 2 d , we get: ( a + 5 d ) − ( a + 2 d ) = 12 3 d = 12 Dividing both sides by 3, we find the common difference: d = 3 12 = 4
Finding the First Term Now that we have the common difference d = 4 , we can substitute it into the equation 2 a + 19 d = 42 to find the first term a :
2 a + 19 ( 4 ) = 42 2 a + 76 = 42 2 a = 42 − 76 2 a = − 34 a = 2 − 34 = − 17
Finding the 20th Term Finally, we can find the 20th term using the formula a 20 = a + 19 d :
a 20 = − 17 + 19 ( 4 ) a 20 = − 17 + 76 a 20 = 59
Conclusion Therefore, the common difference is 4, the first term is -17, and the 20th term is 59.
Examples
Arithmetic progressions are useful in various real-life scenarios, such as calculating simple interest, predicting salary increases, and modeling the depreciation of assets. For instance, if you deposit a fixed amount into a savings account each month, the total amount in your account over time forms an arithmetic progression. Understanding APs can help you estimate future savings or plan investments effectively. Another example is in theater seating arrangements, where the number of seats in each row increases by a constant amount, forming an AP.
The common difference (d) is 4, the first term (a) is 17, and the 20th term is 93 in the arithmetic progression.
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