9. The 5th and 10th terms of an arithmetic sequence are 23 and 48 respectively.
a) Write down the general term of the arithmetic sequence.
b) Find the common difference and first term of the sequence.
c) Is the 15th term of the sequence 74?
Answer (1)
Determine the general term of the arithmetic sequence using a n = a 1 + ( n − 1 ) d .
Find the common difference d and the first term a 1 by solving the system of equations derived from the given information.
Calculate the 15th term a 15 using the general term formula.
Conclude whether the 15th term is 74 or not. The general term is a n = 5 n − 2 , d = 5 , a 1 = 3 , and the 15th term is not 74. a n = 5 n − 2 , d = 5 , a 1 = 3 , No
Explanation
Understanding the Problem We are given an arithmetic sequence where the 5th term is 23 and the 10th term is 48. Our goal is to find the general term of the sequence, the common difference and the first term, and to determine if the 15th term is 74.
General Term Formula Let a n represent the n th term of the arithmetic sequence, a 1 be the first term, and d be the common difference. The general term of an arithmetic sequence is given by the formula: a n = a 1 + ( n − 1 ) d
Setting up Equations We know that a 5 = 23 and a 10 = 48 . Using the general term formula, we can write two equations: a 5 = a 1 + ( 5 − 1 ) d = a 1 + 4 d = 23 a 10 = a 1 + ( 10 − 1 ) d = a 1 + 9 d = 48
Finding the Common Difference Now we have a system of two equations with two variables, a 1 and d :
a 1 + 4 d = 23 a 1 + 9 d = 48 We can solve this system by subtracting the first equation from the second equation to eliminate a 1 :
( a 1 + 9 d ) − ( a 1 + 4 d ) = 48 − 23 5 d = 25 d = 5
Finding the First Term Now that we have found the common difference d = 5 , we can substitute it back into one of the equations to solve for a 1 . Let's use the first equation: a 1 + 4 d = 23 a 1 + 4 ( 5 ) = 23 a 1 + 20 = 23 a 1 = 3
General Term of the Sequence Now that we have found a 1 = 3 and d = 5 , we can write the general term of the arithmetic sequence: a n = a 1 + ( n − 1 ) d a n = 3 + ( n − 1 ) 5 a n = 3 + 5 n − 5 a n = 5 n − 2
Finding the 15th Term To determine if the 15th term is 74, we substitute n = 15 into the general term formula: a 15 = 5 ( 15 ) − 2 a 15 = 75 − 2 a 15 = 73
Conclusion Since a 15 = 73 , the 15th term of the sequence is not 74.
Final Answer a) The general term of the arithmetic sequence is a n = 5 n − 2 .
b) The common difference is d = 5 and the first term is a 1 = 3 .
c) The 15th term of the sequence is not 74, it is 73.
Examples
Arithmetic sequences are used in various real-life scenarios, such as calculating simple interest, predicting patterns, and determining the number of seats in an auditorium. For example, if you deposit a fixed amount of money into a savings account each month, the total amount in your account over time forms an arithmetic sequence. Understanding arithmetic sequences helps in financial planning and forecasting.
