sienna2266
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I get how to do this question
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But was just wondering.. Isn’t ak = (-2)^(k-1) the kth term of a geometric sequence because we’re multiplying by -2 each time?
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Hammad(214508)
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Yes you are right
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sienna2266
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(Original post by Hammad(214508))
Yes you are right
Thanks very much!
Also, I get how to do this question as well.
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But was just wondering.. Isn’t ak=2^k the kth term of a geometric sequence because we’re multiplying by 2 each time?

For ak=2^k to be the kth term of a geometric sequence also doesn’t make sense though, and I have explained this below:

a0=1
a1 =2
a2 =4
a3 =8
a4 =16

We are multiplying by 2 each time. The first term (a1 =2 not a0 =1) is 2.

Using the formula ak =ark-1, the kth term of the geometric sequence is ak =2 x 2k-1. This doesn’t make sense as the kth term mentioned in the question for this sequence is ak=2^k..
Could you please help me understand this?
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Hammad(214508)
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(Original post by sienna2266)
Thanks very much!
Also, I get how to do this question as well.
Name:  3.png
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Attachment 720500720504

But was just wondering.. Isn’t ak=2^k the kth term of a geometric sequence because we’re multiplying by 2 each time?

For ak=2^k to be the kth term of a geometric sequence also doesn’t make sense though, and I have explained this below:

a0=1a1 =2a2 =4
a3 =8
a4 =16

We are multiplying by 2 each time. The first term (a1 =2 not a0 =1) is 2.

Using the formula ak =ark-1, the kth term of the geometric sequence is ak =2 x 2k-1. This doesn’t make sense as the kth term mentioned in the question for this sequence is ak=2^k..
Could you please help me understand this?
Look at the number below the sigma notation, the first one start with 1, while the second one starts at 0 (0 to 4)
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sienna2266
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(Original post by Hammad(214508))
Look at the number below the sigma notation, the first one start with 1, while the second one starts at 1 (1 to 4)
Sorry but I am still confused
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RDKGames
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(Original post by sienna2266)
Using the formula ak =ark-1, the kth term of the geometric sequence is ak =2 x 2k-1. This doesn’t make sense as the kth term mentioned in the question for this sequence is ak=2^k..
Could you please help me understand this?
Don't understand your question TBH.

Of a geo sequence, ar^{k} does not specify the kth term, it specifies the (k+1)th term. But ar^{k-1} does specify the kth term, and note that 2 \cdot 2^{k-1} = 2^k
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sienna2266
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(Original post by RDKGames)
Don't understand your question TBH.

Of a geo sequence, ar^{k} does not specify the kth term, it specifies the (k+1)th term. But ar^{k-1} does specify the kth term, and note that 2 \cdot 2^{k-1} = 2^k
Ahh this makes sense cause 2 x2^(k-1) is the same thing as 2^k!

Just to double check:
2iii) is a geometric series, right?
And the first term of this geometric series is a1=2 and not a0=1 right?

1(iv) is a geometric series as well?
And the first term of this geometric series is a1=1?

So a1 is pretty much the first term for any geometric/arithmetic series/sequence?

Many thanks
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RDKGames
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(Original post by sienna2266)
Ahh this makes sense cause 2 x2^(k-1) is the same thing as 2^k!

Just to double check:
2iii) is a geometric series, right?
And the first term of this geometric series is a1=2 and not a0=1 right?
It's a geometric series of the sequence a_k = 2^{k-1} with a_1=1.

(Original post by sienna2266)
1(iv) is a geometric series as well?
And the first term of this geometric series is a1=1?
It's a geometric series of a sequence a_k = (-2)^{k-1} with a_1=1

(Original post by sienna2266)
So a1 is pretty much the first term for any geometric/arithmetic series/sequence?
Yes, you want a_1 to correspond to the 1st term of your sequence. Wouldn't make much sense to refer to the '0th' term
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sienna2266
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(Original post by RDKGames)
It's a geometric series of the sequence a_k = 2^{k-1} with a_1=1.



It's a geometric series of a sequence a_k = (-2)^{k-1} with a_1=1



Yes, you want a_1 to correspond to the 1st term of your sequence. Wouldn't make much sense to refer to the '0th' term
Thanks, but isn't 2iii) ak = 2^k or ak=2 x 2^(k-1)? Also, shouldn't a1 = 2?
I have just attached it here again:
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RDKGames
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(Original post by sienna2266)
Thanks, but isn't 2iii) ak = 2^k or ak=2 x 2^(k-1)? Also, shouldn't a1 = 2?
I have just attached it here again:
If it starts at 2, then you're completely ignoring the actual first term of 1.

Note that \displaystyle \sum_{k=0}^{k=4} 2^k = \sum_{k+1=1}^{k+1=5} 2^{(k+1)-1} then map (k+1) \mapsto k and we have \displaystyle \sum_{k=1}^5 2^{k-1} which is the sum of the sequence \{ 2^{k-1} \} from the first term up to the 5th term.
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sienna2266
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(Original post by RDKGames)
If it starts at 2, then you're completely ignoring the actual first term of 1.

Note that \displaystyle \sum_{k=0}^{k=4} 2^k = \sum_{k+1=1}^{k+1=5} 2^{(k+1)-1} then map (k+1) \mapsto k and we have \displaystyle \sum_{k=1}^5 2^{k-1} which is the sum of the sequence \{ 2^{k-1} \} from the first term up to the 5th term.
Thanks, I think I understand your explanation but am unsure why you have to go through a conversion process from 2^k to 2^k-1. I've outlined what I am confused with below:

If you are given
a0=1
a1 =2
a2 =4
a3 =8
a4 =16
and asked to write out the sequence formula for this...
The first term is a0 which is 1 (this doesn't sound right because a1 should be the first term but then 1 is the first term in the sequence so I am confused here).

The kth term of the sequence using the formula ak = ar^(k-1) and the fact that first term=a0=1, is ak=1 x 2^(k-1) --> ak= 2^(k-1).
However, in the question, ak= 2^k from the sequence a0 to a4.

Instead, if we take the first term as a1 which is 2 (this also doesn't sound right because 2 is the second term in the sequence).

The kth of the sequence using the formula ak= ar^(k-1) and the fact that first term=a1=2,
is ak= 2 x 2^(k-1) which is the same thing as ak= 2^k. And in the question, ak= 2^k.

Could you please help me with this? Many thanks
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RDKGames
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(Original post by sienna2266)
...
In that case the formula is just a_k=2^k if you start from a_0 as the initial term.
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sienna2266
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(Original post by RDKGames)
In that case the formula is just a_k=2^k if you start from a_0 as the initial term.
But the thing is if you start from a0 as the initial term, the kth term of the sequence is ak=2^(k-1).

This is shown here:
r=2 and a0=1
Sub them both into the formula ak=ar^(k-1) and you get ak=2^(k-1) and not ak=2^k.

In the picture I have attached here, ak=2^k where a0=1,a1=2,a2=4,a3=8 and a4=16
so why is it only when we sub a1=2 into ak=ar^(k-1), that we get ak=2^k?
As you said, "the formula is just Image if you start from Image as the initial term" but this is not the case here.
Would really appreciate if you could please let me know what you think. Many thanks.
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RDKGames
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(Original post by sienna2266)
But the thing is if you start from a0 as the initial term, the kth term of the sequence is ak=2^(k-1).

This is shown here:
r=2 and a0=1
Sub them both into the formula ak=ar^(k-1) and you get ak=2^(k-1) and not ak=2^k.

In the picture I have attached here, ak=2^k where a0=1,a1=2,a2=4,a3=8 and a4=16
so why is it only when we sub a1=2 into ak=ar^(k-1), that we get ak=2^k?
As you said, "the formula is just Image if you start from Image as the initial term" but this is not the case here.
Would really appreciate if you could please let me know what you think. Many thanks.
TBH you're starting to confuse even me because I'm lost as to what your issue with this is or what you're trying to get at.

Look, \displaystyle \sum_{k=0}^{4} 2^k is just a different way to write the sum of the first 5 terms of the sequence \{ 2^{k-1} \} for k=1,2,3,.... This sequence does not carry the notion of having a 0th term because this sequence starts with a_1=1, a_2 = 2, a_3 = 4, ....

If you decide to allow the 0th term as a thing, then the sequence is \{ 2^k \} for k=0,1,2,... so that a_0=1, a_1=2, ... and we can say that the kth term here is a_k=2^k but we have to start with 0th term. So when you say 2nd term, you're really referring to the 3rd one down the line.
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sienna2266
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(Original post by RDKGames)
If you decide to allow the 0th term as a thing, then the sequence is \{ 2^k \} for k=0,1,2,... so that a_0=1, a_1=2, ... and we can say that the kth term here is a_k=2^k but we have to start with 0th term. So when you say 2nd term, you're really referring to the 3rd one down the line.
Ohhh! the last line answered my question. 😅😂 Thank you so much!!
So to get ak=2^k from the sequence 1,2,4,4,16 you sub in r=2 and and a1=2 (cause the first term is 2 and 0th term is 1) into ak=ar^(k-1).
Please kindly let me know about my progress😅
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