Video: CBSE Class X • Pack 5 • 2014 • Question 10

CBSE Class X • Pack 5 • 2014 • Question 10

03:04

Video Transcript

If the first and last terms of an arithmetic progression are five and 45, respectively, and the sum of all its terms is 400, find its common difference.

Remember, an arithmetic progression is a sequence of numbers such that the difference between each of the consecutive terms is constant. We have two formulae which we can apply to arithmetic progressions. The first is the formula for the 𝑛th term. It’s given by 𝑎 plus 𝑛 minus one multiplied by 𝑑, where 𝑎 is the first term and 𝑑 is the common difference. The second is the formula for the sum of the first 𝑛 terms in the sequence. It’s 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one 𝑑.

For a finite sequence such as ours, this is sometimes written as 𝑛 over two multiplied by 𝑎 plus 𝑙, where 𝑙 is the last term in the sequence. We can use this second version to help us calculate the number of terms in our sequence. Since the first term is five and the last is 45, we can substitute five and 45 into this formula, in place of 𝑎 and 𝑙. We also know that the sum of these terms is 400. So we can equate this expression to 400 to get 400 equals 𝑛 over two multiplied by five plus 45. Five plus 45 is 50. And we can solve this equation for 𝑛 by first dividing everything by 50. 40 divided by five is eight. So similarly, 400 divided by 50 is also eight.

Since we want to calculate the value of 𝑛, we can next multiply everything by two. And that gives us 𝑛 is equal to 16. We now know that there are 16 terms in our sequence. Now that we know there are 16 terms, we can use this information to work out the common difference.

Let’s now refer back to the first formula, the one that helps us find any term in the sequence. We know that the 16th term now is 45. And we still have the first term of five. Substituting these into the formula for the 𝑛th term gives us 45 is equal to five plus 16 minus one 𝑑. 16 minus one is 15. So we get 45 equals five plus 15𝑑. To solve for 𝑑, we’ll subtract five from both sides. And that gives us 40 is equal to 15𝑑. Next, we’ll divide through by 15. 𝑑 is therefore equal to 40 over 15.

Finally, we’ll need to simplify this fraction by dividing both the numerator and the denominator by their common factor, which is five. 40 divided by five is eight. And 15 divided by five is three. So we have a common difference of eight-thirds.

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