3,9,27,81,243.... to the 15th term
You can see in the sequence that the previous term is timesed by 3 to produce the next term, for example 3x3=9 (term 1 x 3 to produce term 2).
Also, as three is the first term, then we can see the sequence as powers of three:
term 1 = 3^1
term 2 = 3^2
term 3 = 3^3
and so on....
So the power corresponds to the number of the term. So the 15th term will be 3^15.
Using Alison's method, we can sum together the first 15 terms of the sequence. If we call the original sequence 's' (standing for the sum of the sequence), then times it by 3 we get 3s:
3s= 3^2,3^3,3^4,3^5... This will cause our 15th term to be 3^16.
If we take s from 3s, the two cancel each other out so we get 2s=3^16 -3
To gain s, we need to divide through by 2 to get:
s = (3^16 -3)/2
This gives us the sum of the first fifteen terms of the sequence.
5,10,20,40,80....
In this sequence we times the previous term by 2 to gain the next (5x2 = 10)
so s= 5,5x2,5x2^3,5x2^4,5x2^5....
The 12th term would therefore be 5x2^11
In other words, the sequence is 2.5x2^i, i being the numebr of the term.
If we times the sequence by 2 as in ALison's method, we get:
2s= 5x2,5x2^2,5x2^3,5x2^4,5x2^5..... 12th term being 5x2^12
If we take s from 2s, we gain s= 5x2^12 -5
This is the sum of the sequence.
(3x2^(i-1))
In the symbol for sum, we can tell from the twenty above that the first 20 terms of the sequence are wanted. the i tells us that the sequence begins from the first term.
According to the number of the term (i) we can substitute it into the expression (3x2^(i-1)), this gives us the sequence:
3,6,12,24,48....
From this we can see the previous term is doubled each time to gain the next. As the first term is 3, the overall expression for the sequence is 3x2^n.
In other words:
3,3x2,3x2^2,3x2^3,3x2^4....
This will be s, the twentieth term of which will be 3x2^19.
If we times by 2 to gain 2s, then we have:
3x2,3x2^2,3x2^3,3x2^4,3x2^5.....the twentieth term of which is 3x2^20.
If we take s from 2s, then we get:
s= 3x2^20 -3
The sum of the first 20 terms.
1/2,1/4,1/8,1/16,1/32......
Fromt this we can see the sequence is being divided by 2 each time.
THis ,means the denominator of the fraction is being timesed by 2.
So the expression for the sequence is 1/2^n, n being the number of the term:
2^1 =2
2^2=4
The tenth term will be 1/2^10, which is 1/1024.
If we call the orignal sequence s, then we time it by to to get 2s, we get:
2s= 1,1/2,1/4,1/8,1/16,1/32..... Tenth term 1/2^9 or 1/512
If we take s from 2s,the terms cancel out and we get:
s= 1- 1/1024
This is the sum of the first 10 terms of the progression.
a+ar+ar^2+ar^3......ar^n-1
To get to the next term in this sequence we times by r each time,the power of r is decided by taking 1 from the numebr of the term,so axr^(i-1)
If we use the origianl as s, then times it by r to gain rs, then we get:
rs= ar,ar^2,ar^3,ar^4.....ar^n
As in this sequence the power of r is the number of the term.
If we take rs from s we get:
rs -s = ar^n -1
It is rs -s because we do not no the value of r, so we can't take s. To solve this we factorise, to get:
s(r-1)
Then divide the other side by this to get:
s= (ar^n -1)/(r-1)
The sum of the terms up to the nth term.