Question

Find the sum to \(n\) terms of the series \(1+(1+2)+(1+2+3)+(1+2+3+4)+\ldots\) (a) \(\frac{n(n+1)(2 n+1)}{6}\) (b) \(\frac{n(n+1)(n+2)}{6}\) (c) \(\frac{n(n+1)(n+2)}{12}\) (d) \(\frac{n(n+1)}{2}\)

Short answer

Question: Determine the sum of the series \(1+(1+2)+(1+2+3)+(1+2+3+4)+\ldots\) up to \(n\) terms. Answer: The sum to \(n\) terms of the given series is \(\frac{n(n+1)(n+2)}{6}\).

Step-by-step solution

Sum of first k natural numbers

We need to find the sum of the first \(k\) natural numbers. The formula for this is given by \(\frac{k(k+1)}{2}\).

Sum of the series up to n terms

Let's call the sum of the series up to \(n\) terms as \(S_n\). Now, we need to sum the series in the form of \(\frac{k(k+1)}{2}\), where \(k\) goes from \(1\) to \(n\). So, \(S_n = \sum_{k=1}^{n} \frac{k(k+1)}{2}\).

Simplify the summation

We can simplify the summation as follows: \(S_n = \frac{1}{2}\sum_{k=1}^{n}k(k+1) = \frac{1}{2}\sum_{k=1}^{n}(k^2+k)\). Now, using the properties of summation, we can split this into two separate summations: \(S_n = \frac{1}{2}\left(\sum_{k=1}^{n}k^2+\sum_{k=1}^{n}k\right)\).

Simplify the summations

We know the following formulas for the sum of natural numbers and the sum of the square of natural numbers: \(\sum_{k=1}^{n}k = \frac{n(n+1)}{2}\) and \(\sum_{k=1}^{n}k^2 = \frac{n(n+1)(2n+1)}{6}\). Using these formulas, we can substitute the summations in \(S_n\): \(S_n = \frac{1}{2}\left(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}\right)\).

Simplify the expression

Now, we can simplify the expression for \(S_n\) further: $S_n = \frac{1}{2}\left(\frac{n(n+1)(2n+1)}{3}+n(n+1)\right) = \frac{n(n+1)}{6}(2n+1+3) = \frac{n(n+1)(2n+4)}{6} = \frac{n(n+1)(n+2)}{6}$. Comparing this result with the given options, we see that it matches option (b). So, the sum to \(n\) terms of the given series is \(\frac{n(n+1)(n+2)}{6}\).

Key concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference of any two successive members is a constant, known as the common difference. For those new to the topic, imagine you’re climbing stairs and every step you take is the same height—that constant height is like the common difference in an arithmetic series.

An example of an arithmetic series could be like the sequence 3, 6, 9, 12, where each number is increased by 3, the common difference. However, the series in our exercise is a bit different—it’s a sum of the first n natural numbers. This is where the sum fluctuates with every term, adding yet another layer, as if adding an extra step to our current stair with each move.

Summation Formulas

Grasping summation formulas is akin to learning the shortcuts on a calculator; they simplify the process enormously. For quantitative tasks that involve adding series of numbers, knowing these formulas can turn a daunting mountain of calculations into a manageable molehill.

Let's consider two significant summation formulas featured in our exercise: \(\sum_{k=1}^{n}k = \frac{n(n+1)}{2}\) and \(\sum_{k=1}^{n}k^2 = \frac{n(n+1)(2n+1)}{6}\). Both of these are invaluable for anyone in the field of mathematics, physics, engineering, or any discipline involving quantitative analysis. They serve as the basic building blocks in solving more complex problems. Understanding these shortcuts ensures efficiency and accuracy in calculations.

Quantitative Aptitude

Quantitative aptitude encompasses the ability to handle numbers and undertake numerical calculations effortlessly. It’s not just about knowing the math, but also about applying it in the most effective way. For students and professionals alike, this skill is essential. It's like the know-how of math—applying logic and number sense to solve problems efficiently and accurately.

What brought us here was a problem grounded in quantitative aptitude—finding the sum of a given number series. This involves using reasoning skills and knowledge of formulas to dissect the problem into manageable pieces. By recognizing the pattern of the problem and using the right technique, the student not only finds the solution but also hones their quantitative aptitude, setting a solid foundation for tackling more complex problems in their academic and professional lives.