If $k+7, 2k-2$ and $2k+6$ are three consecutive terms of an A.P., then the value of $k$ is:
(A) 15
(B) 17
(C) 5
(D) 1
Answer (2)
Recognize that in an arithmetic progression, the difference between consecutive terms is constant.
Set up the equation ( 2 k − 2 ) − ( k + 7 ) = ( 2 k + 6 ) − ( 2 k − 2 ) .
Simplify both sides of the equation to get k − 9 = 8 .
Solve for k to find k = 17 .
Explanation
Understanding the Problem We are given that k + 7 , 2 k − 2 , and 2 k + 6 are three consecutive terms of an arithmetic progression (A.P.). Our goal is to find the value of k . In an arithmetic progression, the difference between consecutive terms is constant. This means that the difference between the second term and the first term is equal to the difference between the third term and the second term.
Setting up the Equation We can set up the equation: ( 2 k − 2 ) − ( k + 7 ) = ( 2 k + 6 ) − ( 2 k − 2 ) . Now, let's simplify the equation.
Simplifying the Equation First, simplify the left side of the equation: ( 2 k − 2 ) − ( k + 7 ) = 2 k − 2 − k − 7 = k − 9 Next, simplify the right side of the equation: ( 2 k + 6 ) − ( 2 k − 2 ) = 2 k + 6 − 2 k + 2 = 8 So, we have the equation: k − 9 = 8
Solving for k Now, we solve for k by adding 9 to both sides of the equation: k = 8 + 9 = 17 Therefore, the value of k is 17.
Final Answer Thus, the value of k is 17 .
Examples
Arithmetic progressions are useful in various real-life scenarios, such as calculating simple interest, predicting patterns, and optimizing resource allocation. For instance, 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 progression. Understanding A.P. helps in financial planning and forecasting.
The value of k is 17 . This was determined by setting the differences between the terms of the arithmetic progression equal to each other and solving the resulting equation. Hence, the correct answer is 17 .
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