Find the sum of the numbers in the first n brackets

[pencile]

Find the total number of terms in the 1st n brackets of the series
(1) + (2 + 2^2) + (2^3 + 2^4 + 2^5) + (2^6 + 2^7 + 2^8 +2^9) + ...
where the rth bracket contains r terms.
Hence, find the sum of the numbers in the first n brackets.


Total number of terms in the 1st n brackets
= 1 + 2 + 3 + ... + n = n/sum/r=1 (r) = 1/2 n (n + 1)

Sum of the numbers in the first n brackets
= ?/sum/r=0 (2^r)

I'm not sure how to proceed from here. Thanks for helping.

 

[ksdhart2]

Well, we can start by looking at the individual terms of the sequence and see if we can find a pattern there.

\(\displaystyle a_0 = 2^0 = 1\)
\(\displaystyle a_1 = 2^1 + 2^2 = 2 + 4 = 6\)
\(\displaystyle a_2 = 2^3 + 2^4 + 2^5 = 8 + 16 + 32 = 56\)
\(\displaystyle a_3 = 2^6 + 2^7 + 2^8 + 2^9 = 64 + 128 + 256 + 512 = 960\)

Hm... this doesn't seem to be going anywhere. But the problem asks about the sum of the first n terms, so let's investigate that.

\(\displaystyle \displaystyle \sum_{i=0}^{0} \: a_i = 2^0 = 1\)
\(\displaystyle \displaystyle \sum_{i=0}^{1} \: a_i = 2^0 + 2^1 + 2^2 = 1 + 2 + 4 = 7\)
\(\displaystyle \displaystyle \sum_{i=0}^{2} \: a_i = 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 = 1 + 2 + 4 + 8 + 16 + 32 = 63\)
\(\displaystyle \displaystyle \sum_{i=0}^{3} \: a_i = 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 = 1023\)

Are you seeing a pattern in these terms? Does that pattern help you figure out what the general expression for \(\displaystyle a_i\) is? How does that relate to the sum of the first n terms? Does your solution incorporate knowing how many individual numbers there are in the first n terms?

 

[pencile]

Well, we can start by looking at the individual terms of the sequence and see if we can find a pattern there.

\(\displaystyle a_0 = 2^0 = 1\)
\(\displaystyle a_1 = 2^1 + 2^2 = 2 + 4 = 6\)
\(\displaystyle a_2 = 2^3 + 2^4 + 2^5 = 8 + 16 + 32 = 56\)
\(\displaystyle a_3 = 2^6 + 2^7 + 2^8 + 2^9 = 64 + 128 + 256 + 512 = 960\)

Hm... this doesn't seem to be going anywhere. But the problem asks about the sum of the first n terms, so let's investigate that.

\(\displaystyle \displaystyle \sum_{i=0}^{0} \: a_i = 2^0 = 1\)
\(\displaystyle \displaystyle \sum_{i=0}^{1} \: a_i = 2^0 + 2^1 + 2^2 = 1 + 2 + 4 = 7\)
\(\displaystyle \displaystyle \sum_{i=0}^{2} \: a_i = 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 = 1 + 2 + 4 + 8 + 16 + 32 = 63\)
\(\displaystyle \displaystyle \sum_{i=0}^{3} \: a_i = 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 = 1023\)

Are you seeing a pattern in these terms? Does that pattern help you figure out what the general expression for \(\displaystyle a_i\) is? How does that relate to the sum of the first n terms? Does your solution incorporate knowing how many individual numbers there are in the first n terms?

I can't figure out the expression for a_i and the way the values of i correspond to the number of brackets instead of terms.


Geometric series = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + ...
Formula for each term = (1) 2^r-1
Sum of terms = [(1/2)(n)(n+1)]/sum/r=1 (2^r-1)
= [(1) (2^{(1/2)(n)(n+1)} - 1)] / (2 - 1)
= 2^{(1/2)(n)(n+1)} - 1

It's less confusing this way but I'm not sure if it's correct. It doesn't feel right to slip in the formula for sum of geometric series like that.

Is that an actual classroom problem or
something you made up?
Asking because it is not an arithmetic
or a geometric series.

It's from a textbook, not modified in any way.

 

[ksdhart2]

I can't figure out the expression for a_i and the way the values of i correspond to the number of brackets instead of terms.


Geometric series = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + ...
Formula for each term = (1) 2^r-1
Sum of terms = [(1/2)(n)(n+1)]/sum/r=1 (2^r-1)
= [(1) (2^{(1/2)(n)(n+1)} - 1)] / (2 - 1)
= 2^{(1/2)(n)(n+1)} - 1

It's less confusing this way but I'm not sure if it's correct. It doesn't feel right to slip in the formula for sum of geometric series like that.

It's from a textbook, not modified in any way.

Ah, yes. I misspoke earlier, sorry. You got what I was hinting at though, so that's good. Your answer's absolutely correct. The series as given isn't directly a geometric series, but it can be manipulated to be written as one, as you've seen.