A.S. and G.S. thanks

1.
(i)
The number of layers "from top to bottom" form an arithmeticsequence.
The first term, T(1) = a
d = 40 - 36 = 36 - 32 = 4

T(9) = 40
a + 8d = 40
a + 8 × 4 = 40
a = 8
No. of barrels in the top layer = 8

(ii)
Total number of barrels, S(9)
= n [T(1) + T(n)] / 2
= 9 × (8 + 40) / 2
= 216

(iii)
No. of barrels in the upmost 4 layers, S(4)
= n [2a + (n - 1)d] / 2
= 4 × [2 × 8 + 3 × 4] / 2
= 56


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3.
The annual income in 2007, T(1) = a = $96000
Common ratio, r = 1 + 0.8% = 1.08

Total number of terms from 2007 to 2014
= 2014 - 2007 + 1
= 8

Total earnings from 2007 to 2014,
= a (rⁿ - 1) / (r - 1)
= $96000 × (1.08⁸ - 1) / (1.08 - 1)
= $1021116


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4.
Total vertical distance that the ball move, S(∞)
= 2 × (3.5 + 3.5 × 75% + 3.5 × 75%² + 3.5× 75%³ + ......) m
= 2 × 3.5 / (1 - 75%) m
= 28 m