IB AI · Unit 2 · 2A Arithmetic Sequences
Pick the sum formula from the information you already have. A summation in symbol form can go straight into the calculator.
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SL 1.2
\(u_n = u_1 + (n-1)d\)
\(S_n = \dfrac{n}{2}\bigl(2u_1 + (n-1)d\bigr)\)
\(S_n = \dfrac{n}{2}(u_1 + u_n)\)
The sum of n terms
If you already know the first term and the last term, use
\(S_n = \dfrac{n}{2}(u_1 + u_n)\)
If you know the first term and the common difference, use
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
The first term and the common difference are given.
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(u_1 = 6\)
\(d = 4\)
\(n = 10\)
\(S_{10} = \dfrac{10}{2}\left(2 \times 6 + (10-1) \times 4\right)\)
\(S_{10} = \dfrac{10}{2}\left(12 + 9 \times 4\right)\)
\(S_{10} = \dfrac{10}{2}\left(12 + 36\right)\)
\(S_{10} = \dfrac{10}{2} \times 48\)
\(S_{10} = 5 \times 48\)
\(S_{10} = 240\)
Answer: the sum = \(240\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(u_1 = 5\)
\(d = 3\)
\(n = 8\)
\(S_{8} = \dfrac{8}{2}\left(2 \times 5 + (8-1) \times 3\right)\)
\(S_{8} = \dfrac{8}{2}\left(10 + 7 \times 3\right)\)
\(S_{8} = \dfrac{8}{2}\left(10 + 21\right)\)
\(S_{8} = \dfrac{8}{2} \times 31\)
\(S_{8} = 4 \times 31\)
\(S_{8} = 124\)
Answer: the sum = \(124\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(u_1 = 4\)
\(d = 6\)
\(n = 12\)
\(S_{12} = \dfrac{12}{2}\left(2 \times 4 + (12-1) \times 6\right)\)
\(S_{12} = \dfrac{12}{2}\left(8 + 11 \times 6\right)\)
\(S_{12} = \dfrac{12}{2}\left(8 + 66\right)\)
\(S_{12} = \dfrac{12}{2} \times 74\)
\(S_{12} = 6 \times 74\)
\(S_{12} = 444\)
Answer: the sum = \(444\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(u_1 = 10\)
\(d = 2\)
\(n = 15\)
\(S_{15} = \dfrac{15}{2}\left(2 \times 10 + (15-1) \times 2\right)\)
\(S_{15} = \dfrac{15}{2}\left(20 + 14 \times 2\right)\)
\(S_{15} = \dfrac{15}{2}\left(20 + 28\right)\)
\(S_{15} = \dfrac{15}{2} \times 48\)
\(S_{15} = 15 \times 24\)
\(S_{15} = 360\)
Answer: the sum = \(360\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(u_1 = 7\)
\(d = 5\)
\(n = 9\)
\(S_{9} = \dfrac{9}{2}\left(2 \times 7 + (9-1) \times 5\right)\)
\(S_{9} = \dfrac{9}{2}\left(14 + 8 \times 5\right)\)
\(S_{9} = \dfrac{9}{2}\left(14 + 40\right)\)
\(S_{9} = \dfrac{9}{2} \times 54\)
\(S_{9} = 9 \times 27\)
\(S_{9} = 243\)
Answer: the sum = \(243\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(u_1 = 20\)
\(d = -3\)
\(n = 10\)
\(S_{10} = \dfrac{10}{2}\left(2 \times 20 + (10-1) \times \left(-3\right)\right)\)
\(S_{10} = \dfrac{10}{2}\left(40 + 9 \times \left(-3\right)\right)\)
\(S_{10} = \dfrac{10}{2}\left(40 - 27\right)\)
\(S_{10} = \dfrac{10}{2} \times 13\)
\(S_{10} = 5 \times 13\)
\(S_{10} = 65\)
Answer: the sum = \(65\)
The first term and the last term are given.
\(S_n = \dfrac{n}{2}(u_1 + u_n)\)
\(u_1 = 4\)
\(u_n = 40\)
\(n = 10\)
\(S_{10} = \dfrac{10}{2}\left(4 + 40\right)\)
\(4 + 40 = 44\)
\(S_{10} = \dfrac{10}{2} \times 44\)
\(S_{10} = 5 \times 44\)
\(S_{10} = 220\)
Answer: the sum = \(220\)
\(S_n = \dfrac{n}{2}(u_1 + u_n)\)
\(u_1 = 3\)
\(u_n = 39\)
\(n = 10\)
\(S_{10} = \dfrac{10}{2}\left(3 + 39\right)\)
\(3 + 39 = 42\)
\(S_{10} = \dfrac{10}{2} \times 42\)
\(S_{10} = 5 \times 42\)
\(S_{10} = 210\)
Answer: the sum = \(210\)
\(S_n = \dfrac{n}{2}(u_1 + u_n)\)
\(u_1 = 8\)
\(u_n = 68\)
\(n = 16\)
\(S_{16} = \dfrac{16}{2}\left(8 + 68\right)\)
\(8 + 68 = 76\)
\(S_{16} = \dfrac{16}{2} \times 76\)
\(S_{16} = 8 \times 76\)
\(S_{16} = 608\)
Answer: the sum = \(608\)
\(S_n = \dfrac{n}{2}(u_1 + u_n)\)
\(u_1 = 15\)
\(u_n = 3\)
\(n = 7\)
\(S_{7} = \dfrac{7}{2}\left(15 + 3\right)\)
\(15 + 3 = 18\)
\(S_{7} = \dfrac{7}{2} \times 18\)
\(S_{7} = 7 \times 9\)
\(S_{7} = 63\)
Answer: the sum = \(63\)
\(S_n = \dfrac{n}{2}(u_1 + u_n)\)
\(u_1 = 2\)
\(u_n = 50\)
\(n = 13\)
\(S_{13} = \dfrac{13}{2}\left(2 + 50\right)\)
\(2 + 50 = 52\)
\(S_{13} = \dfrac{13}{2} \times 52\)
\(S_{13} = 13 \times 26\)
\(S_{13} = 338\)
Answer: the sum = \(338\)
\(S_n = \dfrac{n}{2}(u_1 + u_n)\)
\(u_1 = 11\)
\(u_n = 71\)
\(n = 21\)
\(S_{21} = \dfrac{21}{2}\left(11 + 71\right)\)
\(11 + 71 = 82\)
\(S_{21} = \dfrac{21}{2} \times 82\)
\(S_{21} = 21 \times 41\)
\(S_{21} = 861\)
Answer: the sum = \(861\)
Copy the summation into a TI-84 Plus CE.
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(2x+7\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 8.
7. Press ENTER.
The calculator shows \(128\).
Answer: the sum = \(128\)
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(3x+1\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 10.
7. Press ENTER.
The calculator shows \(175\).
Answer: the sum = \(175\)
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(5x-2\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 12.
7. Press ENTER.
The calculator shows \(366\).
Answer: the sum = \(366\)
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(4x+3\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 6.
7. Press ENTER.
The calculator shows \(102\).
Answer: the sum = \(102\)
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(2x+5\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 15.
7. Press ENTER.
The calculator shows \(315\).
Answer: the sum = \(315\)
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(x+8\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 9.
7. Press ENTER.
The calculator shows \(117\).
Answer: the sum = \(117\)
Find the missing first term or common difference, then the sum.
The second term is one common difference after the first term.
\(d = 9 - 5\)
\(d = 4\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(S_{10} = \dfrac{10}{2}\left(2 \times 5 + (10-1) \times 4\right)\)
\(S_{10} = \dfrac{10}{2}\left(10 + 9 \times 4\right)\)
\(S_{10} = \dfrac{10}{2}\left(10 + 36\right)\)
\(S_{10} = \dfrac{10}{2} \times 46\)
\(S_{10} = 5 \times 46\)
\(S_{10} = 230\)
Answer: the sum = \(230\)
The second term is one common difference after the first term.
\(d = 11 - 6\)
\(d = 5\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(S_{8} = \dfrac{8}{2}\left(2 \times 6 + (8-1) \times 5\right)\)
\(S_{8} = \dfrac{8}{2}\left(12 + 7 \times 5\right)\)
\(S_{8} = \dfrac{8}{2}\left(12 + 35\right)\)
\(S_{8} = \dfrac{8}{2} \times 47\)
\(S_{8} = 4 \times 47\)
\(S_{8} = 188\)
Answer: the sum = \(188\)
The second term is one common difference after the first term.
\(d = 7 - 4\)
\(d = 3\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(S_{12} = \dfrac{12}{2}\left(2 \times 4 + (12-1) \times 3\right)\)
\(S_{12} = \dfrac{12}{2}\left(8 + 11 \times 3\right)\)
\(S_{12} = \dfrac{12}{2}\left(8 + 33\right)\)
\(S_{12} = \dfrac{12}{2} \times 41\)
\(S_{12} = 6 \times 41\)
\(S_{12} = 246\)
Answer: the sum = \(246\)
The second term is one common difference after the first term.
\(d = 16 - 20\)
\(d = -4\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(S_{9} = \dfrac{9}{2}\left(2 \times 20 + (9-1) \times \left(-4\right)\right)\)
\(S_{9} = \dfrac{9}{2}\left(40 + 8 \times \left(-4\right)\right)\)
\(S_{9} = \dfrac{9}{2}\left(40 - 32\right)\)
\(S_{9} = \dfrac{9}{2} \times 8\)
\(S_{9} = 9 \times 4\)
\(S_{9} = 36\)
Answer: the sum = \(36\)
The second term is the first term plus one common difference.
\(u_2 = u_1 + d\)
\(14 = u_1 + 3\)
\(u_1 = 14 - 3\)
\(u_1 = 11\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(S_{10} = \dfrac{10}{2}\left(2 \times 11 + (10-1) \times 3\right)\)
\(S_{10} = \dfrac{10}{2}\left(22 + 9 \times 3\right)\)
\(S_{10} = \dfrac{10}{2}\left(22 + 27\right)\)
\(S_{10} = \dfrac{10}{2} \times 49\)
\(S_{10} = 5 \times 49\)
\(S_{10} = 245\)
Answer: the sum = \(245\)
The second term is the first term plus one common difference.
\(u_2 = u_1 + d\)
\(8 = u_1 + 5\)
\(u_1 = 8 - 5\)
\(u_1 = 3\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(S_{7} = \dfrac{7}{2}\left(2 \times 3 + (7-1) \times 5\right)\)
\(S_{7} = \dfrac{7}{2}\left(6 + 6 \times 5\right)\)
\(S_{7} = \dfrac{7}{2}\left(6 + 30\right)\)
\(S_{7} = \dfrac{7}{2} \times 36\)
\(S_{7} = 7 \times 18\)
\(S_{7} = 126\)
Answer: the sum = \(126\)
The second term is the first term plus one common difference.
\(u_2 = u_1 + d\)
\(18 = u_1 + \left(-2\right)\)
\(u_1 = 18 - \left(-2\right)\)
\(u_1 = 20\)
\(S_n = \dfrac{n}{2}\left(2u_1 + (n-1)d\right)\)
\(S_{12} = \dfrac{12}{2}\left(2 \times 20 + (12-1) \times \left(-2\right)\right)\)
\(S_{12} = \dfrac{12}{2}\left(40 + 11 \times \left(-2\right)\right)\)
\(S_{12} = \dfrac{12}{2}\left(40 - 22\right)\)
\(S_{12} = \dfrac{12}{2} \times 18\)
\(S_{12} = 6 \times 18\)
\(S_{12} = 108\)
Answer: the sum = \(108\)
If you can write a formula for uₙ, the summation key adds the first n terms.
\(u_n = 4n + 1\)
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(4x + 1\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 12.
7. Press ENTER.
The calculator shows \(324\).
Answer: the sum = \(324\)
\(u_n = 3n + 2\)
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(3x + 2\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 20.
7. Press ENTER.
The calculator shows \(670\).
Answer: the sum = \(670\)
\(u_n = 5n - 1\)
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(5x - 1\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 15.
7. Press ENTER.
The calculator shows \(585\).
Answer: the sum = \(585\)
\(u_n = u_1 + (n-1)d\)
\(u_1 = 7\)
\(d = 4\)
\(u_n = 7 + (n-1) \times 4\)
\(u_n = 7 + 4n - 4\)
\(u_n = 4n + 3\)
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(4x + 3\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 10.
7. Press ENTER.
The calculator shows \(250\).
Answer: the formula is \(u_n = 4n + 3\) and the sum = \(250\)
\(d = 11 - 6\)
\(d = 5\)
\(u_n = u_1 + (n-1)d\)
\(u_1 = 6\)
\(d = 5\)
\(u_n = 6 + (n-1) \times 5\)
\(u_n = 6 + 5n - 5\)
\(u_n = 5n + 1\)
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(5x + 1\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 8.
7. Press ENTER.
The calculator shows \(188\).
Answer: the formula is \(u_n = 5n + 1\) and the sum = \(188\)
\(u_1 = 9 - 3\)
\(u_1 = 6\)
\(u_n = u_1 + (n-1)d\)
\(u_1 = 6\)
\(d = 3\)
\(u_n = 6 + (n-1) \times 3\)
\(u_n = 6 + 3n - 3\)
\(u_n = 3n + 3\)
On a TI-84 Plus CE:
1. Press MATH.
2. Press 0 to open summation Σ(.
3. Type the expression \(3x + 3\).
4. Move to the variable box and type \(x\).
5. Move to the lower limit and type 1.
6. Move to the upper limit and type 11.
7. Press ENTER.
The calculator shows \(231\).
Answer: the formula is \(u_n = 3n + 3\) and the sum = \(231\)
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