Sequences and series

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1. Sigma notation and arithmetic series

Warm-up

Three quick questions from earlier lessons.

Discover

To add 1+2+⋯+1001 + 2 + \dots + 100, pair the first and last terms: 1+1001 + 100, 2+992 + 99, … How many pairs, and what is each pair worth?

  1. 50 pairs of 101
  2. 100 pairs of 101

Key idea

∑k=pqak\displaystyle\sum_{k=p}^{q} a_k means ap+ap+1+⋯+aqa_p + a_{p+1} + \dots + a_q: it has q−p+1q - p + 1 terms.

The sum of the nn terms of an arithmetic sequence is Sn=n(a1+an)2S_n = \dfrac{n(a_1 + a_n)}{2}: the number of terms times the average of the first and last. In particular 1+2+⋯+n=n(n+1)21 + 2 + \dots + n = \dfrac{n(n + 1)}{2}.

Worked example

Find ∑k=120(3k+2)\displaystyle\sum_{k=1}^{20} (3k + 2).

First term 55, last term 6262
k = 1 and k = 20.
S=20(5+62)2S = \dfrac{20(5 + 62)}{2}
20 terms.
S=670S = 670
10 · 67.

Your turn, guided

Your turn: 2+4+6+⋯+402 + 4 + 6 + \dots + 40.

n=20n = 20
40 ÷ 2: 20 terms.
S=420S = 420
20 · 42 / 2.

Practice ★

  1. How many terms does ∑k=317(2k+1)\displaystyle\sum_{k=3}^{ 17 } (2k + 1) have?
  2. How many terms does ∑k=321(2k+1)\displaystyle\sum_{k=3}^{ 21 } (2k + 1) have?
  3. How many terms does ∑k=238(2k+1)\displaystyle\sum_{k=2}^{ 38 } (2k + 1) have?

Watch out!

Practice ★★

  1. Find ∑k=130(3k+7)\displaystyle\sum_{k=1}^{ 30 } (3k + 7).
  2. Find ∑k=111(6k+8)\displaystyle\sum_{k=1}^{ 11 } (6k + 8).
  3. Find ∑k=121(6k+6)\displaystyle\sum_{k=1}^{ 21 } (6k + 6).
  1. Find 1+2+3+⋯+941 + 2 + 3 + \dots + 94.
  2. Find 1+2+3+⋯+801 + 2 + 3 + \dots + 80.

Exit ticket

Three questions to check you have got it.

  1. Find ∑k=130(3k+7)\displaystyle\sum_{k=1}^{ 30 } (3k + 7).
  2. Find ∑k=111(6k+8)\displaystyle\sum_{k=1}^{ 11 } (6k + 8).
  3. Find ∑k=121(6k+6)\displaystyle\sum_{k=1}^{ 21 } (6k + 6).

Summary card

Arithmetic series

  • ∑k=pq\displaystyle\sum_{k=p}^{q} has q−p+1q - p + 1 terms.
  • Sn=n(a1+an)2S_n = \dfrac{n(a_1 + a_n)}{2}.
  • 1+2+⋯+n=n(n+1)21 + 2 + \dots + n = \dfrac{n(n + 1)}{2}.

Watch out: count the terms with +1+1.

2. Finite geometric series

Warm-up

Three quick questions from earlier lessons.

Discover

S=1+2+4+8+16S = 1 + 2 + 4 + 8 + 16. What is 2S−S2S - S?

  1. 32−1=3132 - 1 = 31
  2. 1616

Key idea

Multiplying by rr and subtracting gives the sum of a geometric sequence.

The sum of the first nn terms of a geometric sequence (r≠1r \ne 1) is Sn=a1 rn−1r−1S_n = a_1\,\dfrac{r^n - 1}{r - 1}.

Worked example

Find 3+6+12+…3 + 6 + 12 + \dots (8 terms).

S8=3⋅28−12−1S_8 = 3 \cdot \dfrac{2^8 - 1}{2 - 1}
a₁ = 3, r = 2, n = 8.
S8=3⋅255=765S_8 = 3 \cdot 255 = 765
2⁸ = 256.

Your turn, guided

Your turn: 1+3+9+…1 + 3 + 9 + \dots (5 terms).

S5=35−13−1S_5 = \dfrac{3^5 - 1}{3 - 1}
a₁ = 1, r = 3.
S5=121S_5 = 121
(243 − 1)/2.

Practice ★

  1. Find the sum of the first 6 terms of the geometric sequence with a1=8a_1 = 8 and ratio 3.
  2. Find the sum of the first 5 terms of the geometric sequence with a1=7a_1 = 7 and ratio 2.
  3. Find the sum of the first 5 terms of the geometric sequence with a1=4a_1 = 4 and ratio 2.

Watch out!

Practice ★★

  1. Find ∑k=062k\displaystyle\sum_{k=0}^{ 6 } 2^k.
  2. Find ∑k=0102k\displaystyle\sum_{k=0}^{ 10 } 2^k.
  1. You put 764 dollars in a savings account at the end of each year, at 5 % interest a year. How much is in the account right after the 12th payment? Round to the nearest cent.
  2. You put 344 dollars in a savings account at the end of each year, at 5 % interest a year. How much is in the account right after the 11th payment? Round to the nearest cent.

Exit ticket

Three questions to check you have got it.

  1. Find the sum of the first 6 terms of the geometric sequence with a1=8a_1 = 8 and ratio 3.
  2. Find the sum of the first 5 terms of the geometric sequence with a1=7a_1 = 7 and ratio 2.
  3. Find the sum of the first 5 terms of the geometric sequence with a1=4a_1 = 4 and ratio 2.

Summary card

Finite geometric series

  • Sn=a1 rn−1r−1S_n = a_1\,\dfrac{r^n - 1}{r - 1}.
  • nn is the number of terms; the power is rnr^n.

Watch out: the last term is a1r n−1a_1 r^{\,n-1}, but the sum uses rnr^n.

Answers: Sequences and series

1. Sigma notation and arithmetic series

A

  1. 1515
  2. 1919
  3. 3737

B

  1. 16051605
  2. 484484
  3. 15121512

C

  1. 44654465
  2. 32403240

D

  1. 16051605
  2. 484484
  3. 15121512

2. Finite geometric series

A

  1. 29122912
  2. 217217
  3. 124124

B

  1. 127127
  2. 20472047

C

  1. 62262705467484843151200000000000\frac{622627054674848431}{51200000000000}
  2. 62555325233155031280000000000\frac{6255532523315503}{1280000000000}

D

  1. 29122912
  2. 217217
  3. 124124