Indexing with sigma notation

Sigma notation is a compact way to represent series, and in the HSC Mathematics Extension 1 course you’ll often need to convert it into an algebraic expression to solve for the number of terms. This problem blends series formulas with quadratic equations and forces you to interpret the discriminant carefully — a great test of whether a solution exists within the integers. As we work through it, you’ll see how to decompose a sum, apply standard summation formulas and then reason about the nature of the solutions. Mastering this process will build your confidence in algebraic manipulation and series analysis.

Problem Statement

Given

$$\sum_{k=1}^{n}(3k-1)=119,$$

find $n$.

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Hints

Split the sum into $3\sum k - \sum 1$.

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Solutions

We split the sum using the hint:

$$\sum_{k=1}^{n}(3k-1)=3\sum_{k=1}^{n}k-\sum_{k=1}^{n}1.$$

To evaluate these separate sums, we use well-known summation formulas. The sum of the first $n$ natural numbers is $\frac{n(n+1)}{2}$, and adding $1$ exactly $n$ times gives $n$. Substituting these in yields:

$$\sum_{k=1}^{n}(3k-1)=3\cdot\frac{n(n+1)}{2}-n=\frac{3n^2+n}{2}.$$

This simplified expression represents the left-hand side of the given equation. We are told this sum equals $119$, so we set the expression equal to $119$ and rearrange:

$$\frac{3n^2+n}{2}=119 \implies 3n^2+n-238=0.$$

To check whether an integer $n$ can satisfy the equation, we compute the discriminant. A perfect square discriminant indicates rational roots, while a non-perfect square means the solutions are irrational. Examine the discriminant:

$$\Delta=1+2856=2857.$$

This is not a perfect square, so the quadratic has irrational roots. Hence there is no integer $n$ for which the original sum equals exactly $119$.

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Takeaways

• Splitting a sum into separate parts lets you use simple summation formulas like the sum of natural numbers.
• Always check whether the solution to the resulting equation lies in the natural numbers; a non‑perfect‑square discriminant rules out integer $n$.
• Interpreting sigma notation as an algebraic expression is essential for solving many HSC sequence and series problems.

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Further Readings

HSC Polys Ext 1, HSC Combinatorics, HSC Vectors, HSC Trigonometry