Exercise 8.1 answers: Sequences and Series

Open NCERT p. 138Matches NCERT’s answer

1. Given: aₙ = n(n + 2).
2. The rule says: multiply the position number by (that number + 2).
3. n = 1: 1 × 3 = 3
4. n = 2: 2 × 4 = 8
5. n = 3: 3 × 5 = 15
6. n = 4: 4 × 6 = 24
7. n = 5: 5 × 7 = 35

Answer3, 8, 15, 24, 35

Watch this explained “The position and the term”, 1:08 into A rule that turns a position number into a term

Open NCERT p. 138Matches NCERT’s answer

1. Put n = 1, 2, 3, 4, 5 into aₙ = n/(n + 1), keeping each answer as an exact fraction.
2. n = 1: 1/2
3. n = 2: 2/3
4. n = 3: 3/4
5. n = 4: 4/5
6. n = 5: 5/6

Answer1/2, 2/3, 3/4, 4/5, 5/6

Watch this explained “The position and the term”, 1:08 into A rule that turns a position number into a term

Open NCERT p. 138Matches NCERT’s answer

1. Put n = 1, 2, 3, 4, 5 into aₙ = 2ⁿ.
2. n = 1: 2¹ = 2
3. n = 2: 2² = 4
4. n = 3: 2³ = 8
5. n = 4: 2⁴ = 16
6. n = 5: 2⁵ = 32

Answer2, 4, 8, 16, 32

Watch this explained “The position and the term”, 1:08 into A rule that turns a position number into a term

Open NCERT p. 138Matches NCERT’s answer

1. Put n = 1, 2, 3, 4, 5 into aₙ = (2n − 3)/6, keeping each answer as an exact fraction.
2. n = 1: (2 − 3)/6 = −1/6
3. n = 2: (4 − 3)/6 = 1/6
4. n = 3: (6 − 3)/6 = 1/2
5. n = 4: (8 − 3)/6 = 5/6
6. n = 5: (10 − 3)/6 = 7/6

Answer−1/6, 1/6, 1/2, 5/6, 7/6

Watch this explained “The position and the term”, 1:08 into A rule that turns a position number into a term

Open NCERT p. 138Matches NCERT’s answer

1. The power of (−1) flips the sign every time n goes up by 1; the power of 5 grows.
2. n = 1: (−1)⁰ × 5² = 25
3. n = 2: (−1)¹ × 5³ = −125
4. n = 3: (−1)² × 5⁴ = 625
5. n = 4: (−1)³ × 5⁵ = −3125
6. n = 5: (−1)⁴ × 5⁶ = 15625

Answer25, −125, 625, −3125, 15625

Watch this explained “Where the signs catch you”, 10:39 into A rule that turns a position number into a term

Open NCERT p. 138Matches NCERT’s answer

1. Put n = 1, 2, 3, 4, 5 into aₙ = n(n² + 5)/4, keeping each answer as an exact fraction.
2. n = 1: 1 × 6 / 4 = 3/2
3. n = 2: 2 × 9 / 4 = 9/2
4. n = 3: 3 × 14 / 4 = 21/2
5. n = 4: 4 × 21 / 4 = 21
6. n = 5: 5 × 30 / 4 = 75/2

Answer3/2, 9/2, 21/2, 21, 75/2

Watch this explained “The position and the term”, 1:08 into A rule that turns a position number into a term

Open NCERT p. 138Matches NCERT’s answer

1. Put the wanted position number into aₙ = 4n − 3.
2. a₁₇ = 4 × 17 − 3 = 68 − 3 = 65
3. a₂₄ = 4 × 24 − 3 = 96 − 3 = 93

Answera₁₇ = 65, a₂₄ = 93

Watch this explained “The position and the term”, 1:08 into A rule that turns a position number into a term

Open NCERT p. 138Matches NCERT’s answer

1. Put n = 7 into aₙ = n²/2ⁿ.
2. a₇ = 7²/2⁷ = 49/128

Answera₇ = 49/128

Watch this explained “The position and the term”, 1:08 into A rule that turns a position number into a term

Open NCERT p. 138Matches NCERT’s answer

1. Put n = 9 into aₙ = (−1)ⁿ⁻¹ n³.
2. n − 1 = 8 is even, so (−1)⁸ = 1.
3. a₉ = 1 × 9³ = 729

Answera₉ = 729

Watch this explained “Where the signs catch you”, 10:39 into A rule that turns a position number into a term

Open NCERT p. 138Matches NCERT’s answer

1. Given: aₙ = n(n − 2)/(n + 3); find a₂₀.
2. Put n = 20 into aₙ = n(n − 2)/(n + 3).
3. a₂₀ = 20 × 18 / 23 = 360/23

Answera₂₀ = 360/23

Watch this explained “The position and the term”, 1:08 into A rule that turns a position number into a term

Open NCERT p. 139Matches NCERT’s answer

1. Here each term is built from the one before it: triple it and add 2.
2. a₁ = 3
3. a₂ = 3 × 3 + 2 = 11
4. a₃ = 3 × 11 + 2 = 35
5. a₄ = 3 × 35 + 2 = 107
6. a₅ = 3 × 107 + 2 = 323
7. The corresponding series just adds these terms one after another: 3 + 11 + 35 + 107 + 323 + …

AnswerFirst five terms: 3, 11, 35, 107, 323. Series: 3 + 11 + 35 + 107 + 323 + …

Watch this explained “When the rule looks back”, 5:13 into A rule that turns a position number into a term

Open NCERT p. 139Matches NCERT’s answer

1. Here each term is the one before it, divided by the new position number.
2. a₁ = −1
3. a₂ = −1/2
4. a₃ = (−1/2)/3 = −1/6
5. a₄ = (−1/6)/4 = −1/24
6. a₅ = (−1/24)/5 = −1/120
7. The corresponding series adds these terms: −1 + (−1/2) + (−1/6) + (−1/24) + (−1/120) + …

AnswerFirst five terms: −1, −1/2, −1/6, −1/24, −1/120. Series: −1 − 1/2 − 1/6 − 1/24 − 1/120 − …

Watch this explained “Terms first, series second”, 7:13 into What changes once the terms are added instead of listed

Open NCERT p. 139Matches NCERT’s answer

1. The first two terms are both 2. From the third term on, each term is 1 less than the one before it.
2. a₁ = 2
3. a₂ = 2
4. a₃ = 2 − 1 = 1
5. a₄ = 1 − 1 = 0
6. a₅ = 0 − 1 = −1
7. The corresponding series adds these terms: 2 + 2 + 1 + 0 + (−1) + …

AnswerFirst five terms: 2, 2, 1, 0, −1. Series: 2 + 2 + 1 + 0 − 1 + …

Watch this explained “Terms first, series second”, 7:13 into What changes once the terms are added instead of listed

Open NCERT p. 139Matches NCERT’s answer

1. Build the Fibonacci terms first: each term is the sum of the two before it.
2. a₁ = 1, a₂ = 1, a₃ = 2, a₄ = 3, a₅ = 5, a₆ = 8
3. n = 1: a₂/a₁ = 1/1 = 1
4. n = 2: a₃/a₂ = 2/1 = 2
5. n = 3: a₄/a₃ = 3/2
6. n = 4: a₅/a₄ = 5/3
7. n = 5: a₆/a₅ = 8/5

Answer1, 2, 3/2, 5/3, 8/5

Watch this explained “When the rule looks back”, 5:13 into A rule that turns a position number into a term