If $3, x, y, 24$ are in AP, find $x+y$.
Answer (1)
The given sequence 3 , x , y , 24 is an arithmetic progression.
Determine the common difference d using the equation 24 = 3 + 3 d , which gives d = 7 .
Calculate x and y using x = 3 + d and y = 3 + 2 d , resulting in x = 10 and y = 17 .
Find the sum x + y = 10 + 17 , which equals 27 .
Explanation
Understanding Arithmetic Progression We are given that 3 , x , y , 24 are in arithmetic progression (AP). This means that the difference between consecutive terms is constant. Let's denote this common difference by d .
Expressing Terms with Common Difference We can express x , y , and 24 in terms of the first term 3 and the common difference d as follows:
x = 3 + d y = 3 + 2 d 24 = 3 + 3 d
Solving for the Common Difference Now, we can solve for d using the equation 24 = 3 + 3 d :
24 − 3 = 3 d 21 = 3 d d = 3 21 d = 7
Finding x and y Now that we have the value of d , we can find the values of x and y :
x = 3 + d = 3 + 7 = 10 y = 3 + 2 d = 3 + 2 ( 7 ) = 3 + 14 = 17
Calculating x + y Finally, we can find the sum of x and y :
x + y = 10 + 17 = 27
Final Answer Therefore, the value of x + y is 27.
Examples
Arithmetic progressions are useful in various real-life scenarios, such as calculating simple interest on a loan or predicting the number of seats in subsequent rows of a stadium. For example, if you deposit $3 into a savings account and the interest adds $7 each year, the amounts in your account each year will form an arithmetic progression: $3, $10, $17, $24, and so on. Understanding arithmetic progressions helps in predicting future values based on a constant rate of change.
