Arithmetic Series: CMAT Exam Notes
Overview
An Arithmetic Series is the sum of terms in an Arithmetic Progression (AP). In an AP, the difference between consecutive terms is constant, known as the common difference ($d$). Recognizing these sequences and knowing how to sum them is a fundamental skill for CMAT Quantitative Ability.
Essential Formulas
To master arithmetic series, you must memorize these two foundational formulas:
- n-th Term ($a_n$): To find any specific term in the sequence: $a_n = a + (n - 1)d$
- Sum of n Terms ($S_n$): To find the sum of the first $n$ terms: $S_n = \frac{n}{2}[2a + (n - 1)d]$
- Alternative Sum Formula: If the last term ($l$ or $a_n$) is known: $S_n = \frac{n}{2}(a + l)$
In these formulas:
- $a$ = First term
- $d$ = Common difference
- $n$ = Number of terms
- $l$ = Last term
Worked Example
Problem: Find the sum of the first 10 terms of the series: $3, 7, 11, 15, ...$
- Identify the variables: The first term $a = 3$. The common difference $d = 7 - 3 = 4$. The number of terms $n = 10$.
- Apply the formula: Use the $S_n$ formula: $S_{10} = \frac{10}{2}[2(3) + (10 - 1)4]$
- Calculate: $S_{10} = 5[6 + (9 \times 4)]$ $S_{10} = 5[6 + 36]$ $S_{10} = 5[42] = 210$
Exam Tips & Traps
- Watch for signs: A negative common difference ($d < 0$) is common in decreasing sequences. Ensure your subtraction ($a_2 - a_1$) preserves the sign.
- Read carefully: Differentiate between finding the '$n^{th}$ term' (which term is this?) versus the 'sum of $n$ terms' (what is the total value?).
- Mental Shortcut: For an arithmetic series, the average of the first and last term multiplied by the number of terms always yields the sum: $S_n = n \times \text{average}$.
Practise these as timed MCQs to lock the pattern in.