Question Video: Expanding a Finite Series | Nagwa Question Video: Expanding a Finite Series | Nagwa

Question Video: Expanding a Finite Series Mathematics • Second Year of Secondary School

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Expand ∑_(𝑟 = 7) ^(11) (−2 + (7/𝑟)).

02:25

Video Transcript

Expand the sum from 𝑟 equals seven to 11 of negative two plus seven over 𝑟.

This is the Greek letter ∑. And when we read this, we say the sum from 𝑟 equals seven to 11. So, given this information, how do we find the sum from 𝑟 equals seven to 11 of the expression negative two plus seven over 𝑟? Well, we’re going to let 𝑟 be equal to seven. We’ll then let it be equal to eight, nine, 10, and 11, and we’ll find the sum of those expressions.

So when 𝑟 is equal to seven, the expression negative two plus seven over 𝑟 is negative two plus seven over seven. And that’s negative two plus one, which is simply negative one. So when 𝑟 is seven, the expression negative two plus seven over 𝑟 is negative one. Let’s repeat this with 𝑟 equals eight. The expression becomes negative two plus seven-eighths. To evaluate this, let’s write negative two with a denominator of eight. It’s negative sixteen eighths. Then we can add the numerators. And we find that when 𝑟 is equal to eight, our expression is negative nine-eighths.

Let’s now work out the value of negative two plus seven over 𝑟 when 𝑟 is equal to nine. We get negative two plus seven-ninths. Negative two with a denominator of nine is negative 18 over nine. And then we see that the expression is equivalent to negative eleven ninths. Next, we let 𝑟 equal 10, and we get negative two plus seven-tenths. We can then write that as negative 20 over 10 plus seven-tenths, which gives us negative thirteen tenths. In a similar way, when we let 𝑟 equal 11, we get negative 15 over 11.

And, of course, the question asks us to expand our sum. So we simply need to add all of these individual expressions. When we do, we get negative one plus negative nine-eighths plus negative 11 over nine plus negative 13 over 10 plus negative 15 over 11. And of course adding a negative is the same as subtracting a positive, so we can rewrite this further.

When we expand our expression then, we get negative one minus nine-eighths minus eleven ninths minus thirteen tenths minus fifteen elevenths.

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