# Solving Positive Integer Problems: First & Last Terms in nth Bracket

• al_201314
In summary, a user named Alvin is seeking help with a question involving positive integers and their brackets. Another user named status offers guidance by using the geometric sum formula and the sum of the first n positive integers to find the first and last terms in each bracket and the sum of the first n brackets. Alvin thanks status for their help.
al_201314
Hi everyone,

This is my first post here Anyway I have problems solving this question wonder anyone could help give me some clues as to how to go about it. Here goes:

The positive integers are bracketed as follows,

(1), (2,3), (4,5,6,7), (8,9,10,11,12,13,14,15), ...

No. of integers in the rth bracket is 2^r-1. State stae first term and last term in the nth bracket. Hence, show that the sum of the terms in the first n brackets is 2^n-1[(2^n)]-1].

Thanks very much.. I have absolutely no clue..

alvin

Start by finding how many numbers are in the first n brackets, which is just the sum of the first n powers of 2 (use the geometric sum formula). This will let you know the first and last number in each bracket. Then, to find the sum, use the sum of the first n positive integers, and plug in what you just found for n. If you don't know the sum of the first n integers, the easiest way to find it is to use a telescoping sum:

$$\sum_{k=1}^{n} (k+1)^2 - k^2$$

If you were to write this out term by term, you'd see each term cancels except the first and last. Now that you have this sum, rewrite it as:

$$\sum_{k=1}^{n} k^2+ 2k + 1 - k^2 = \sum_{k=1}^{n} 2k + 1$$

from which the result should follow pretty easily.

Last edited:
I got it.. thanks status.

## 1. What are positive integer problems?

Positive integer problems involve finding solutions to mathematical equations or statements using only whole numbers greater than zero.

## 2. What is the significance of the first and last terms in nth bracket?

In positive integer problems, the first and last terms in nth bracket are often used as indicators of the pattern or sequence in the problem. They can provide insight into the solution and help in finding the missing terms.

## 3. How do you solve positive integer problems?

To solve positive integer problems, you need to first identify the pattern or sequence in the problem. Then, use this pattern to find the missing terms or the solution. You can also use algebraic equations or trial and error to solve these problems.

## 4. Are there any specific strategies for solving positive integer problems?

Yes, there are various strategies that can be used to solve positive integer problems. Some common strategies include using patterns, creating algebraic equations, and using trial and error. It is important to choose a strategy that works best for the specific problem at hand.

## 5. Can positive integer problems be solved using calculators or other tools?

Yes, calculators or other tools can be used to solve positive integer problems. However, it is important to understand the underlying concepts and strategies involved in solving these problems, rather than solely relying on calculators or tools.

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