Number Series

Run a fixed diagnostic ladder on every series instead of staring at it. Step 1: write the differences between consecutive terms. If they are constant it is arithmetic; if the differences of the differences are constant, it is quadratic. Step 2: if differences explode, take RATIOS — constant ratio means geometric. Step 3: if neither, test the four special shapes: squares or cubes with a small offset, primes, alternating two interleaved series, and 'x n then plus or minus k'. Step 4: check the position number itself as a factor. One of these four covers almost every exam series.

• Check in order: constant difference, constant ratio, differences-of-differences, then squares/cubes.

  First try subtracting consecutive terms — if the gap is constant, it is arithmetic. If not, try dividing — a constant ratio means geometric. If neither works, take differences of the differences, which catches quadratic patterns. Last, check if terms are perfect squares or cubes with an offset.

• Alternating series is two series interleaved.

  When a straightforward single rule fails, split the series into odd-position and even-position terms and check each half separately — many 'trick' series are just two simple series woven together.

• 1Arithmetic series: constant difference between consecutive terms, e.g. 3, 7, 11, 15.
• 2Geometric series: constant ratio between consecutive terms, e.g. 2, 6, 18, 54.
• 3Quadratic (second-difference) series: the differences themselves form an arithmetic series, e.g. 1, 2, 5, 10, 17 (differences 1, 3, 5, 7).
• 4Always write out the differences between consecutive terms on the first pass — it turns most series into an immediately visible pattern.

• Arithmetic progression: nth term = a + (n-1)d, and the sum of n terms = n/2 x [2a + (n-1)d].
• Geometric progression: nth term = a x r^(n-1). A rapidly growing series is a signal to test ratios, not differences.
• Keep the first fifteen squares and the first ten cubes memorised — most 'odd one out' series are a square or cube shifted by 1 or 2.
• The standard trap is an alternating series: odd positions follow one rule and even positions another. If a single rule fails, split the series into two before assuming it is complicated.
• In 'find the wrong term' questions, verify the rule on at least three consecutive terms before rejecting a term — one coincidence is not a pattern.

• Do not assume every series is additive. If differences are not constant, test ratios before inventing a more complex rule.

• A Modern Approach to Verbal and Non-Verbal Reasoning — R S Aggarwal
• Analytical Reasoning — M K Pandey

• NCERT textbooks (free PDFs)