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Be Reasonable

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

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Janusz Asked

In y = ax +b when are a, -b/a, b in arithmetic progression. The polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2 and c be in arithmetic progression?

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Summats Clear

Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.

Speedy Summations

Stage: 5 Challenge Level: Challenge Level:1

Why do this problem?

This problem provides an introduction to summing arithmetic series, and allows students to discover for themselves the formulae used to calculate such sums. By seeing a particular case, students can perceive the structure and see where the general method for summing such series comes from.

The problem could be used to introduce $\sum$ notation.

 

Possible approach

You may wish to show the video, in which Alison works out $\sum_{i=1}^{10} i$ in silence, or you may wish to recreate the video for yourself on the board.

Then write up $\sum_{i=1}^{100} i$ and ask students to adapt Alison's method to work it out. Share answers and explanations of how they worked it out.

 

Next, give students the following questions:

$2+4+6+\dots+96+98+100$

$\sum_{k=1}^{20} (4k+12)$

$37+42+47+52+\dots+102+107+112$

 

"Can you adapt the method to work out these three sums? In a while I'm going to give you another question like these and you'll need to be able to work it out efficiently"

While students are working, listen out for useful comments that they make about how to work out such sums generally. Then bring the class together to share answers and methods for the questions they have worked on.

Make up a few questions like those above, and invite students out to the board to work them out 'on the spot', explaining what they do as they go along.

Next, invite students to create a formula from their general thinking:
"Imagine a sequence that starts at $a$ and goes up in equal steps to the $n^{th}$ term which is $l$. Can you use what you did with the numerical examples to create a formula for the sum of the series?"

Give students time to think and discuss in pairs and then share their suggestions.

"What if you were asked to find the sum of the first $n$ terms of the sequence $a, (a+d), (a + 2d), (a + 3d) $ and so on - can you adapt your formula?"

Again, allow some time for discussion before bringing the class together to share what they did.

Finally, "Can you use your formula to work out after how many terms would $17+21+25+\dots$ be greater than $1000$?"

One teacher blogged about her experiences of using this problem; you can read her account here.

Key questions

What can you say about the sum of the first and last, and the second and penultimate terms of an arithmetic sequence?

How do you know these sums of pairs will always be the same?

 

Possible extension

Challenge students to find the sum of all the integers less than $1000$ which are not divisible by $2$ or $3$.

Summats Clear would make a nice extension challenge for students who have found this problem straightforward.

 

Possible support

Slick Summing explores the same content as this problem but introduces new ideas more slowly and does not use $\sum$ notation.