Video: CBSE Class X • Pack 3 • 2016 • Question 2

CBSE Class X • Pack 3 • 2016 • Question 2

02:22

Video Transcript

What value of 𝑘 will make 𝑘 plus nine, two 𝑘 minus one, and two 𝑘 plus seven consecutive terms of an arithmetic progression?

An arithmetic progression is a sequence of numbers such that the difference between any two consecutive terms is constant. Remember, consecutive simply means one after the other. For example, one, three, five, seven is an example of an arithmetic progression. It has a constant common difference of two. And the first two consecutive terms are one and three. This means that the difference between the first two terms, that’s 𝑘 plus nine and two 𝑘 minus one, must be equal to the difference between the second and the third term. That’s two 𝑘 minus one and two 𝑘 plus seven.

Remember, when we find the difference, we subtract one from the other. So the difference between the first two terms is two 𝑘 minus one minus 𝑘 plus nine. And the difference between the second and third term is two 𝑘 plus seven minus two 𝑘 minus one. Let’s simplify each of these expressions.

On the left-hand side, we get two 𝑘 minus one minus 𝑘 minus nine. It’s really important to be careful when multiplying out the second bracket. Technically, what we’re doing is multiplying everything inside this bracket by negative one, which is why we end up with negative nine. A common mistake is to forget this fact and simply write plus nine.

Similarly, on the right-hand side, we’re multiplying the second bracket by negative one. That gives us negative two 𝑘 plus one. When we simplify the left-hand side of this expression, we get 𝑘 minus 10. Since two 𝑘 minus 𝑘 is 𝑘 and negative one minus nine is negative 10. Two 𝑘 minus two 𝑘 is zero. So we end up with eight on the right-hand side.

And we can solve this expression for 𝑘 by adding 10 to both sides. Eight plus 10 is 18. And we can see that the value of 𝑘 that will make the terms consecutive terms of an arithmetic progression is 18.

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