Arithmetic sequence HELP

1.
(i)
T(1) = 9
T(2) = 14
T(3) = 19
T(4) = 24
T(5) = 29

(ii)
T(1) = -1
T(2) = 1
T(3) = 3
T(4) = 5
T(5) = 7


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2.
(i)
Common difference, d
= 12 - 5 = 19 - 12 = 26 - 19
= 7

General term, T(n)
= 5 + (n - 1) × 7
= 7n - 2


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(ii)
Common difference, d
= (-6) - 3 = (-15) - (-6) = (-24) - (-15)
= -9

General term, T(n)
= 3 + (n - 1) × (-9)
= 12 - 9n


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3.
T(1) = a = 95
T(n) = -1
d = 87 - 95 = -8

95 + (n - 1) × (-8) = -1
103 - 8n = -1
8n = 104
n = 13
Number of terms = 13


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4.
Let y be the arithmetic mean.

y - (-30) = (-18) - y
2y = -18 - 30
y = -24
The arithmetic mean = -24


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5.
a = T(1) = 3

Let d be the common difference.
The four arithmetic means are T(2), T(3), T(4) and T(5).

T(6) = 23
3 + (6 - 1) × d = 23
5d = 20
d = 4
3 + 4 = 7
7 + 4 = 11
11 + 4 = 15
15 + 4 = 19

The four arithmetic means 7, 11, 15and 19.


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6.
(i)
a = 5
d = 9 - 5 = 4
n = 24

S(24)
= n [2a + (n - 1)d] / 2
= 24 × [2 × 5 + 23 × 4] / 2
= 1224

(ii)
a = 70
d = 68 - 2 = -2

T(n) = 30
70 + (n - 1) × (-2) = 30
72 - 2n = 30
2n = 42
n = 21

S(21)
= 21 × (70 + 30) /2
= 1050


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7.
(i)
a = 6
d = 13 - 6 = 7

T(15)
= 6 + 14 × 7
= 104

(ii)
T(30)
= 6 + 29 × 7
= 209

(iii)
The sum from the 15th term to the 30th term
= S(30) - S(14)
= {30 × [6 × 2 + 29 × 7] / 2} - {14 × [6 × 2 + 13 × 7] / 2}
= 3225 - 721
= 2504