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 the equation to k − 9 = 8 .
Solve for k to find k = 17 $.
Explanation
Understanding Arithmetic Progressions We are given that k + 7 , 2 k − 2 , and 2 k + 6 are three consecutive terms of an arithmetic progression (A.P.). In an A.P., 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 write this as: ( 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: k − 9 = 8
Solving for k Now, solve for k :
k = 8 + 9 = 17
Final Answer Therefore, 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 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 progression. Understanding A.P. helps in financial planning and forecasting.
The value of k that makes k + 7 , 2 k − 2 , and 2 k + 6 consecutive terms of an arithmetic progression is 17. The calculation involved setting the differences between the terms equal and solving the resulting equation. Therefore, the correct answer is 17.
;
