MAT practice

meliodas89

11

The sum of the first 2n terms of :
1, 1, 2, 1/2 , 4, 1/4, 8, 1/8, 16, 1/16
is?
Please help!

Reply 1

mqb2766

21

Original post by meliodas89

The sum of the first 2n terms of :
1, 1, 2, 1/2 , 4, 1/4, 8, 1/8, 16, 1/16
is?
Please help!

You should recognise both series so
1+2+4+...+2^(n-1)
1+1/2+1/4+...+1/2^(n-1)
If not, theyre geometric. But it/they should be a write down.

In a similar vein, you could sum the first 2, 4, 6, ... terms and simply spot the general pattern

(edited 2 months ago)

Reply 2

Kallisto

Entertainment Forum Helper

22

Original post by meliodas89

The sum of the first 2n terms of :
1, 1, 2, 1/2 , 4, 1/4, 8, 1/8, 16, 1/16
is?
Please help!

There are two series summed up. If you have a closer look at the integers and fraction numbers, what can be seen? right, they are doubled and divided into halves. They change alternatively. That is the pattern. You just need general terms for both.

Spoiler

(edited 2 months ago)

Reply 3

Kallisto

Entertainment Forum Helper

22

Original post by mqb2766

You should recognise both series so
1+2+4+...+2^(n-1)
1+1/2+1/4+...+1/2^(n-1)
If not, theyre geometric. But it/they should be a write down.

In a similar vein, you could sum the first 2, 4, 6, ... terms and simply spot the general pattern

I recommend you to read the rules in maths forums. It is not allowed to share the solutions with students as long as they did not try to work it out by themselves. Embed the general series in spoilers. @Muttley79 would thank you.

Reply 4

mqb2766

21

Original post by Kallisto

I recommend you to read the rules in maths forums. It is not allowed to share the solutions with students as long as they did not try to work it out by themselves. Embed the general series in spoilers. @Muttley79 would thank you.

Dont think I shared the solution?

Its an old (2010) mat multiple choice question and its easy to google the model solution as well as hints/full solution/discussion (underground). However, the visual proofs for both series are well known and question can be done (ans written down) without using the geometric series formulae.

Similarly as its a multiple choice question, you could note the sum is 2 when n=1 and that eliminates 2/4 of the solutions and noting it must grow as 2^n (rather than 2^(2n)) gives the answer without doing much maths.

(edited 2 months ago)

Reply 5

DFranklin

18

Original post by Kallisto

I recommend you to read the rules in maths forums. It is not allowed to share the solutions with students as long as they did not try to work it out by themselves. Embed the general series in spoilers. @Muttley79 would thank you.

Your rebuke is misplaced. The question asks for the sum; mqb has given no more than the slightest hint towards how to find that sum. In particular, there's no requirement to express the series in any kind of

2^{xyz}

form, and frankly, it's not particularly helpful to do so anyhow - except didactically, as mqb used it, to point out "this is what you're actually looking at here".

I'll also point out mqb has made literally thousands of posts helping people on f38 - he is quite aware of the forum rules and complies with them.

Reply 6

DFranklin

18

Original post by mqb2766

Dont think I shared the solution?

Its an old (2010) mat multiple choice question and its easy to google the model solution as well as hints/full solution/discussion (underground). However, the visual proofs for both series are well known and question can be done (ans written down) without using the geometric series formulae.

Similarly as its a multiple choice question, you could note the sum is 2 when n=1 and that eliminates 2/4 of the solutions and noting it must grow as 2^n (rather than 2^(2n)) gives the answer without doing much maths.

Re the last paragraph: I can't be bothered to dig up the paper, but even if you didn't have prospective answers to choose from, for a MAT candidate, just evaluating for n = 1, 2 should have them thinking "the sum is probably ..." and then n = 3 just confirms it. 30 second job, honestly. (To be clear: I'm agreeing not arguing).

(edited 2 months ago)

Reply 7

Kallisto

Entertainment Forum Helper

22

Original post by DFranklin

Your rebuke is misplaced. The question asks for the sum; mqb has given no more than the slightest hint towards how to find that sum. In particular, there's no requirement to express the series in any kind of

2^{xyz}

form, and frankly, it's not particularly helpful to do so anyhow - except didactically, as mqb used it, to point out "this is what you're actually looking at here".

I'll also point out mqb has made literally thousands of posts helping people on f38 - he is quite aware of the forum rules and complies with them.

Okay, I think the best is that I read the rules for myself through again. And I am sorry, if I was wrong.

@mqb2766 nevermind! looks like that your help did not break with the rules. I retract my words.

Reply 8

mqb2766

21

Original post by meliodas89

The sum of the first 2n terms of :
1, 1, 2, 1/2 , 4, 1/4, 8, 1/8, 16, 1/16
is?
Please help!

Just as an aside, both the model solutions (pmt) and underground go down the route strongly hinted at in the question of treating it as two geometric series and using the finite sum formula for each and combining. As Dfranklin notes, a bit of problem solving (multiple choice/little working) is probably the easiest thing to do, then followed by classic/elementary results for the two (finite) subseries
https://www.mathsisfun.com/algebra/infinite-series.html
for instance. The 1+2+4+... is the rice on a chess board or number of binary values or number of cells in an n-dimensional cube of size 2 in each dimension (-1) or ...

For a worked solution its probably easiest to note its a single finite geometic series of length (2n-1) with a=2^(n-1) and r=1/2 (or equivalently the difference of a couple of infinite series). Then its almost a write down ans and there is no need to split the series into two parts as in the other worked solutions. Then +1 to your answer to account for the double 1.

However, it does help to be a bit more descriptive than "please help" in the OP. Similarly, a bit of problem solving can often get you started on a question or better understand what problem youre actually having or in this case actually solve the question.

(edited 2 months ago)

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