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Quantitative Aptitude · Sequence and Series

Special Series and Sigma Notation for CA Foundation

Updated 1 October 2026 · Fact-checked

Special series are the sums 1 + 2 + … + n, 1² + 2² + … + n² and 1³ + 2³ + … + n³. Sigma notation (Σ) writes them compactly. Use n(n+1)/2, n(n+1)(2n+1)/6 and [n(n+1)/2]². Split the given series into these parts, pull out constants, substitute n, and check with a small n.

Understand Special Series and Sigma Notation

A series is the sum of the terms of a sequence. Some series come up so often that their sums have ready-made formulas. The three you must know are the sums of the first n natural numbers, of their squares, and of their cubes.

Sigma notation is shorthand for a long sum. Σ (k = 1 to n) of f(k) means: put k = 1, 2, 3, … up to n into f(k) and add all the results. The letter k is just a counter. The number below Σ is where you start. The number above is where you stop.

The three standard sums are 1 + 2 + … + n = n(n+1)/2, then 1² + 2² + … + n² = n(n+1)(2n+1)/6, and 1³ + 2³ + … + n³ = [n(n+1)/2]². Notice that the sum of cubes is the square of the sum of the natural numbers. This is a handy memory link.

Most exam questions do not hand you these sums directly. They give something like Σ(2k + 3) or Σk(k+1). Your job is to break the expression into pieces of the form k, k², k³ and constants. Sigma is linear, so you can split a sum of terms, and you can take out a constant multiplier. Then apply the right formula to each piece.

A sum that starts at a number other than 1 is handled by subtraction. The sum from k = m to n equals the sum up to n minus the sum up to (m − 1).

Key formulas to remember

Sum of first n natural numbers
Σk = 1 + 2 + 3 + … + n = n(n + 1) ÷ 2
Works for k from 1 to n, where n is a positive integer.
Sum of squares of first n natural numbers
Σk² = 1² + 2² + … + n² = n(n + 1)(2n + 1) ÷ 6
Do not forget the division by 6. The result is always a whole number.
Sum of cubes of first n natural numbers
Σk³ = 1³ + 2³ + … + n³ = [n(n + 1) ÷ 2]²
Equal to (Σk)². Square the whole bracket, not just n.
Sum of a constant
Σc (k = 1 to n) = n × c
You are adding the constant c, n times.
Linearity of sigma
Σ(a·uₖ + b·vₖ) = a·Σuₖ + b·Σvₖ
Constants come out. Sums of terms split. This does not work for products or powers of sums.
Sum from m to n
Σ (k = m to n) f(k) = Σ (k = 1 to n) f(k) − Σ (k = 1 to m − 1) f(k)
Use when the lower limit is not 1.
Sum of first n odd numbers
1 + 3 + 5 + … + (2n − 1) = n²
Follows from Σ(2k − 1) = 2·n(n+1)/2 − n.
Sum of first n even numbers
2 + 4 + 6 + … + 2n = n(n + 1)
Follows from 2·Σk.
Sum of k(k + 1)
1·2 + 2·3 + … + n(n + 1) = n(n + 1)(n + 2) ÷ 3
Derived by splitting into Σk² + Σk. Useful as a check in options.

How to solve Special Series and Sigma Notation questions

Use this method for any question that asks you to find the sum of a series built from natural numbers, their squares or cubes.

  1. 1Read the limits. Note the starting value of k and the last value n.
  2. 2Expand the general term into powers of k. For example, k(k + 1) becomes k² + k.
  3. 3Split the sum into separate sums of k³, k², k and constants. Take out any constant multipliers.
  4. 4Replace each piece with its formula. A constant c summed n times gives n × c.
  5. 5Substitute the value of n and calculate carefully. Cancel common factors before multiplying.
  6. 6If the lower limit is not 1, find the sum to the top limit and subtract the sum up to one less than the lower limit.
  7. 7Verify by putting n = 1 or n = 2 into your answer, or by adding two or three terms by hand, and then match with the options.

Quickest way: Test small n and eliminate options

When to use it: Use when the question gives a general n and four algebraic options, or when n is a small number and the options differ in the last digit or size.

  1. For a formula in n, put n = 1. Compute the series by hand (it is just the first term) and cross out every option that does not give that value.
  2. If options still remain, put n = 2 and add two terms by hand. Cross out mismatches.
  3. For a numeric n, make a rough size estimate first. The sum of squares up to n is about n³ ÷ 3 (a little more), so for n = 10 expect a value a little above 333. This removes options that are far off.
  4. For sum of cubes, find Σk first and just square it. Never expand cubes one by one.
  5. If the lower limit is not 1, subtract two standard sums rather than adding many terms.
  6. Skip the question if it needs a long derivation you cannot start in 30 seconds. A wrong answer costs 0.25 marks.

Common mistakes in Special Series and Sigma Notation

  • Writing Σk² = [n(n+1)/2]² or mixing up the squares and cubes formulas.

    Both formulas look alike and both involve n(n+1).

    Fix: Remember: cubes give a square of the bracket. Squares give three factors over 6. Check with n = 2: squares give 5, cubes give 9.

  • Forgetting to divide by 6 in the sum of squares.

    Students remember the three factors n, (n+1), (2n+1) but drop the denominator.

    Fix: Test n = 1. The formula must give 1. Without the 6 it would give 6.

  • Writing Σc = c instead of n·c.

    The constant has no k, so it feels like nothing is being summed.

    Fix: Σ means add once for each value of k. If k runs from 1 to n, you add c a total of n times.

  • Treating Σk² as (Σk)².

    Students believe squaring can move outside the sum.

    Fix: Sigma only splits over addition. For n = 3, Σk² = 14 but (Σk)² = 36.

  • Using the standard formula directly when the sum starts from a number other than 1.

    The standard formulas are built for k starting at 1.

    Fix: Subtract the sum up to (m − 1) from the sum up to n.

  • Miscounting the number of terms when subtracting, such as using n − m instead of n − m + 1.

    Students count the gap, not the number of terms.

    Fix: Number of terms from m to n is n − m + 1. For a constant summed from 4 to 9, that is 6 terms.

Worked examples

Example 1

The value of 1² + 2² + 3² + … + 10² is: (A) 285 (B) 385 (C) 330 (D) 3025

Show the solution
  1. This is the sum of squares with n = 10.
  2. Use Σk² = n(n + 1)(2n + 1) ÷ 6.
  3. Substitute: 10 × 11 × 21 ÷ 6.
  4. 10 × 11 = 110, and 110 × 21 = 2310.
  5. 2310 ÷ 6 = 385.
  6. Option (D) 3025 is the sum of cubes up to 10, and (A) 285 is the sum of squares up to 9, so these are traps.

Answer: (B) 385

Example 2

The value of Σ (k = 1 to 12) (2k + 3) is: (A) 168 (B) 192 (C) 204 (D) 156

Show the solution
  1. Split: Σ(2k + 3) = 2Σk + Σ3.
  2. Σk for n = 12 is 12 × 13 ÷ 2 = 78.
  3. So 2Σk = 2 × 78 = 156.
  4. Σ3 over 12 terms = 12 × 3 = 36.
  5. Add: 156 + 36 = 192.
  6. Check by hand: the terms are 5, 7, 9, … an AP with 12 terms, first term 5 and last term 27. Sum = 12 × (5 + 27) ÷ 2 = 192.

Answer: (B) 192

Example 3

The value of 11³ + 12³ + 13³ + 14³ + 15³ is: (A) 14400 (B) 3025 (C) 11375 (D) 12375

Show the solution
  1. The series starts at 11, so subtract: Σ (k = 1 to 15) k³ − Σ (k = 1 to 10) k³.
  2. Σk³ up to 15 = [15 × 16 ÷ 2]² = 120² = 14400.
  3. Σk³ up to 10 = [10 × 11 ÷ 2]² = 55² = 3025.
  4. Subtract: 14400 − 3025 = 11375.
  5. Check by adding the cubes directly: 1331 + 1728 + 2197 + 2744 + 3375 = 11375.

Answer: (C) 11375

Exam tips

  • Memorise the three main formulas until you can write them without thinking. Most questions in this topic are direct applications.
  • Always test your answer with n = 1 when options are algebraic. It removes wrong options in seconds.
  • For ranges that do not start at 1, subtract two standard sums. Do not add terms one by one.
  • Look for the setter's traps in the options. Common ones are the sum of cubes in place of squares, or the result without dividing by 6 or 2.
  • Cancel factors before multiplying. For n(n+1)(2n+1) ÷ 6, divide first. It saves time and avoids arithmetic slips.

Practice questions from Sequence and Series

Special Series and Sigma Notation: frequently asked questions

What are the formulas for the sum of n natural numbers, squares and cubes?

The sum of the first n natural numbers is n(n+1)/2. The sum of their squares is n(n+1)(2n+1)/6. The sum of their cubes is [n(n+1)/2]², which is the square of the first sum.

What does sigma notation mean?

Σ means add. Σ (k = 1 to n) f(k) tells you to put k = 1, 2, …, n into f(k) and add all the results. The letter k is only a counter and can be any letter.

How do I find a sum that does not start from 1?

Find the sum up to the top limit and subtract the sum up to one less than the starting value. For example, the sum of squares from 6 to 10 equals Σk² to 10 minus Σk² to 5.

Can I take Σ outside a product or a square?

No. Sigma splits only over addition and subtraction, and constant multipliers can come out. Σk² is not the same as (Σk)², and Σ(k × k+1) must be expanded first.