Summation Formulas for Natural Numbers

Hello! Welcome back to our journey through the CAT quantitative aptitude syllabus.

In our last two lessons, we explored Arithmetic and Geometric Progressions, learning how to find specific terms and, crucially, how to sum a series with a constant difference or a constant ratio. Today, we'll add another powerful set of tools to your problem-solving arsenal. We will focus on three specific, non-negotiable formulas that frequently appear as building blocks in more complex CAT problems.

This lesson directly addresses the learning outcome: Apply formulas for the sum of first n natural numbers, their squares, and their cubes. By the end of our session, you will not only know these formulas by heart but will also be able to manipulate and combine them to solve series problems that don't immediately look like a standard AP or GP.

1. The Three Fundamental Summation Formulas

There are three special series whose sum formulas are essential for any competitive exam. While we derived the formulas for AP and GP sums, these three are best memorized directly. They are the sums of the first $n$ integers, the first $n$ squares, and the first $n$ cubes.

Let's look at them together.

Formulas for Sum of Special Series — This image displays the three core formulas for the sum of the first n natural numbers, their squares, and their cubes.

In mathematical notation, which you may be familiar with from your computer science background (e.g., in algorithm analysis), these are often written using Sigma ( $Σ$ ) notation:

Sum of the first $n$ natural numbers:
$\sum_{i = 1}^{n} i = 1 + 2 + 3 + \dots + n = \frac{n ( n + 1 )}{2}$
Sum of the squares of the first $n$ natural numbers:
$\sum_{i = 1}^{n} i^{2} = 1^{2} + 2^{2} + 3^{2} + \dots + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$
Sum of the cubes of the first $n$ natural numbers:
$\sum_{i = 1}^{n} i^{3} = 1^{3} + 2^{3} + 3^{3} + \dots + n^{3} = [\frac{n ( n + 1 )}{2}]^{2}$

Notice a neat pattern: the formula for the sum of cubes is simply the square of the formula for the sum of natural numbers! Such patterns can make formulas easier to remember under pressure.

To see these formulas in action and build confidence in using them, let's watch a clear explanation for each one.

Summation Formulas and Sigma Notation - Calculus

The video 'Summation Formulas and Sigma Notation' by The Organic Chemistry Tutor provides a concise walkthrough of each of these three formulas with simple, clear examples. We'll watch it in three short parts.

Please watch the following three segments: Sum of n (1:39 - 4:50): Focus on how the formula n(n+1)/2 is applied. Sum of n-squared (10:07 - 12:22): Observe the application of the n(n+1)(2n+1)/6 formula. Sum of n-cubed (12:22 - 13:53): Note the structure of the formula and its relation to the first one.

Test your understanding!

Calculate the sum of the first 10 natural numbers, the sum of their squares, and the sum of their cubes.

Show answer

Here, $n = 10$ .

Sum of numbers: $\frac{10 ( 10 + 1 )}{2} = \frac{10 \times 11}{2} = 5 \times 11 = 55$
Sum of squares: $\frac{10 ( 10 + 1 ) ( 2 \cdot 10 + 1 )}{6} = \frac{10 \times 11 \times 21}{6} = 5 \times 11 \times 7 = 385$
Sum of cubes: $[\frac{10 ( 10 + 1 )}{2}]^{2} = (55)^{2} = 3025$

2. Combining Formulas: Solving Complex Series

CAT questions rarely ask for a direct formula application. Instead, they present a series whose general term is a polynomial in $n$ , and your task is to break it down.

The key lies in two properties of summations:

You can split a sum over addition/subtraction: $\sum (a_{i} \pm b_{i}) = \sum a_{i} \pm \sum b_{i}$
You can pull out a constant factor: $\sum (c \cdot a_{i}) = c \cdot \sum a_{i}$

Example Strategy:
Suppose you need to find the sum of the series $2, 12, 36, 80, \dots$ up to $n$ terms.

Find the general term: Look for a pattern.
- 1st term: $2 = 1^{3} + 1^{2}$
- 2nd term: $12 = 2^{3} + 2^{2}$
- 3rd term: $36 = 3^{3} + 3^{2}$
- 4th term: $80 = 4^{3} + 4^{2}$
  The general $i$ -th term is $i^{3} + i^{2}$ .
Apply the summation: We want to find $\sum_{i = 1}^{n} (i^{3} + i^{2})$ .
Split the sum: $\sum_{i = 1}^{n} i^{3} + \sum_{i = 1}^{n} i^{2}$
Apply the formulas: Substitute the known formulas for sum of cubes and sum of squares and simplify.

This process of identifying the general term, breaking it down, and applying the standard formulas is the core skill for this topic.

Let's watch a demonstration of this technique. The video solves problems like finding the sum of $(i - 1)^{2}$ , which requires expanding the expression first.

Summation Formulas and Sigma Notation - Calculus

We'll continue with the same video. This section shows exactly how to break down more complex expressions into the basic forms we've just learned.

Watch the segment from 13:53 to 20:26. Pay close attention to the strategy: first, perform any algebraic expansion (like foiling (i-1)^2). Second, separate the summation across the plus/minus signs. Finally, apply the three core formulas.

Test your understanding!

Find the sum of the series $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots$ up to 20 terms.

Show answer

Find the general term: The $n$ -th term of the series is $n (n + 1)$ .
Expand the term: $n (n + 1) = n^{2} + n$ .
Apply the summation: We need to find $\sum_{n = 1}^{20} (n^{2} + n)$ .
Split the sum: $\sum_{n = 1}^{20} n^{2} + \sum_{n = 1}^{20} n$ .
Apply the formulas with $n = 20$ :
- $\sum n^{2} = \frac{20 ( 20 + 1 ) ( 2 \cdot 20 + 1 )}{6} = \frac{20 \times 21 \times 41}{6} = 10 \times 7 \times 41 = 2870$ .
- $\sum n = \frac{20 ( 20 + 1 )}{2} = \frac{20 \times 21}{2} = 10 \times 21 = 210$ .
Add the results: $2870 + 210 = 3080$ .

3. Handling Variations and Shortcuts

What if a series doesn't start from 1? For example, find the sum $1 1^{2} + 1 2^{2} + \dots + 2 0^{2}$ .

The strategy is simple: Calculate the sum from 1 to the end, and subtract the sum of the missing part.

$\sum_{i = 11}^{20} i^{2} = (\sum_{i = 1}^{20} i^{2}) - (\sum_{i = 1}^{10} i^{2})$

You would then apply the formula for the sum of squares for $n = 20$ and $n = 10$ and find the difference. This is a very common question pattern.

Finally, for quick reference, it's useful to have all these formulas, including a few others for sums of even and odd numbers, in one place.

CAT QA Number System : Shortcut Methods

The article 'Number System in CAT' from Collegedunia provides a concise list of these summation formulas. It's a perfect 'cheat sheet' for revision.

Read the bullet points under the heading 'CAT QA Number System : Shortcut Methods'. Note the first three formulas we've discussed, and also pay attention to the simple, powerful formulas for the sum of the first n even and odd numbers.

The two additional formulas from that resource are:

Sum of first $n$ odd numbers: $1 + 3 + 5 + \dots = n^{2}$
Sum of first $n$ even numbers: $2 + 4 + 6 + \dots = n (n + 1)$

These can save you significant time if you spot them in a problem.

Conclusion

Today, we've added three indispensable formulas to your toolkit. While AP and GP have flexible formulas based on their first term and common ratio/difference, these special series for sums of numbers, squares, and cubes have fixed formulas that you must memorize. The real skill for the CAT lies in recognizing how to use them as building blocks to solve more elaborate series problems.

Key Takeaways:

Memorize the three core formulas for the sum of the first $n$ natural numbers ( $\sum n$ ), their squares ( $\sum n^{2}$ ), and their cubes ( $\sum n^{3}$ ).
For a complex series, your first step is always to find the general $n$ -th term.
Use algebraic manipulation (e.g., expanding brackets) to express the general term as a sum or difference of powers of $n$ .
Apply the summation formulas to each part of the expanded expression.
If a series starts from a number other than 1, calculate the total sum and subtract the sum of the missing initial terms.

Preview of the Next Lesson:
We will now shift gears from sequences and series to another fundamental area of algebra: logarithms. We will learn what logarithms are, their relationship with exponents, and how to use their properties to simplify complex expressions and solve equations. This is a critical skill for a wide range of quantitative problems.

Recommended resources

1. The Three Fundamental Summation Formulas

2. Combining Formulas: Solving Complex Series

3. Handling Variations and Shortcuts

Conclusion