Sum of an Arithmetic Series

\( S_n=\tfrac{n}{2}\,[2a+(n-1)d]=\tfrac{n}{2}(a_1+a_n) \)
Algebra GCSE
Question 3 of 20
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Find the sum of the first 30 terms of 10, 15, 20,…

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
\( a=10, d=5, n=30. \)

Explanation

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Statement

An arithmetic series is the sum of the terms in an arithmetic sequence, where each term increases by a fixed difference \(d\). The formula for the sum of the first \(n\) terms is:

\[ S_n = \frac{n}{2}\left[2a + (n-1)d\] = \frac{n}{2}(a_1 + a_n) \]

Here, \(a\) (or \(a_1\)) is the first term, \(d\) is the common difference, \(a_n\) is the \(n\)th term, and \(n\) is the number of terms.

Why it’s true

  • Pairing method: Writing the series forwards and backwards shows each pair sums to the same total.
  • For example: \(1+2+3+\cdots+100\). Writing it backwards: \(100+99+98+\cdots+1\). Adding both gives \(101+101+101+\cdots\) (100 times).
  • So, total sum = \(n \times (a_1 + a_n)/2\).

Recipe (how to use it)

  1. Identify the first term \(a\), the common difference \(d\), and the number of terms \(n\).
  2. Either use \(S_n = \tfrac{n}{2}[2a+(n-1)d]\) directly, or find the last term \(a_n\) and use \(S_n = \tfrac{n}{2}(a_1+a_n)\).
  3. Substitute values and simplify.

Spotting it

These appear whenever the question asks for the sum of a sequence with a constant step, e.g. “Find the total of the first 50 even numbers.”

Common pairings

  • Formula for the nth term of an arithmetic sequence: \(a_n = a + (n-1)d\).
  • Word problems involving money, steps, or repeated patterns.

Mini examples

  1. Given: Find the sum of the first 10 terms of \(2, 5, 8, 11, …\). Answer: \(S_{10} = 10/2 × (2+29) = 155\).
  2. Given: Find the sum of the first 20 natural numbers. Answer: \(S_{20} = 20/2 × (1+20) = 210\).

Pitfalls

  • Forgetting to divide by 2 at the end.
  • Mixing up nth term formula with sum formula.
  • Arithmetic slips when simplifying.

Exam strategy

  • Write out first few terms to check the sequence is arithmetic.
  • Use the version of the formula that fits the given information.
  • Always check your answer is reasonable — sums should be much larger than individual terms.

Summary

The sum of an arithmetic series can be found using the formula \(S_n = \frac{n}{2}[2a+(n-1)d]\) or \(S_n = \frac{n}{2}(a_1+a_n)\). Both methods use the idea of pairing terms from the start and end of the sequence.

Worked examples

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  1. Find the sum of the first 10 terms of 3, 7, 11, …
    1. \( a = 3, d = 4, n = 10 \)
    2. \( S_n = 10/2[2(3)+(10-1)4] \)
    3. \( S_n = 5[6+36] = 5×42 \)
    4. \( S_n = 210 \)
    Answer: 210
  2. Find the sum of the first 20 natural numbers.
    1. \( a = 1, d = 1, n = 20 \)
    2. \( S_n = 20/2(1+20) \)
    3. \( S_n = 10×21 \)
    4. \( S_n = 210 \)
    Answer: 210
  3. Find the sum of the first 15 even numbers.
    1. Sequence: 2, 4, 6,…
    2. \( a=2, d=2, n=15 \)
    3. \( S_n = 15/2[2(2)+(14×2)] \)
    4. \( S_n = 15/2[4+28] = 15/2×32 \)
    5. \( S_n = 240 \)
    Answer: 240
  4. Find the sum of the first 12 terms of 5, 10, 15,…
    1. \( a=5, d=5, n=12 \)
    2. \( S_n = 12/2[2(5)+(11×5)] \)
    3. \( S_n = 6[10+55] = 6×65 \)
    4. \( S_n = 390 \)
    Answer: 390
  5. Find the sum of first 30 terms of 1, 4, 7,…
    1. \( a=1, d=3, n=30 \)
    2. \( S_n = 30/2[2(1)+(29×3)] \)
    3. \( S_n = 15[2+87] = 15×89 \)
    4. \( S_n = 1335 \)
    Answer: 1335
  6. Find the sum of the first 25 odd numbers.
    1. Sequence: 1, 3, 5,…
    2. \( a=1, d=2, n=25 \)
    3. \( S_n = 25/2[2(1)+(24×2)] \)
    4. \( S_n = 25/2[2+48] = 25/2×50 \)
    5. \( S_n = 625 \)
    Answer: 625
  7. Find the sum of the first 50 multiples of 6.
    1. \( a=6, d=6, n=50 \)
    2. \( S_n = 50/2[2(6)+(49×6)] \)
    3. \( S_n = 25[12+294] = 25×306 \)
    4. \( S_n = 7650 \)
    Answer: 7650
  8. Find the sum of the first 40 terms of 100, 97, 94,…
    1. \( a=100, d=-3, n=40 \)
    2. \( S_n = 40/2[2(100)+(39×-3)] \)
    3. \( S_n = 20[200-117] = 20×83 \)
    4. \( S_n = 1660 \)
    Answer: 1660
  9. Find the sum of first 60 terms of 7, 14, 21,…
    1. \( a=7, d=7, n=60 \)
    2. \( S_n = 60/2[2(7)+(59×7)] \)
    3. \( S_n = 30[14+413] = 30×427 \)
    4. \( S_n = 12810 \)
    Answer: 12810
  10. Find the sum of first 100 natural numbers.
    1. \( a=1, d=1, n=100 \)
    2. \( S_n = 100/2(1+100) \)
    3. \( S_n = 50×101 \)
    4. \( S_n = 5050 \)
    Answer: 5050