Urgent! Arithmetic sequence

13.
Ifthe sumofthe arithmetic sequence -10, ... , 38 is 182, find thenumber of terms of the sequence.

Solution :
T(1) = -10
T(n) = 38

S(n):
n[T(1) + (T(n)]/2 = 182
n[(-10) + 38]/2 = 182
14n = 182
n = 13

Hence, no. of terms = 13


14.
Howmany consecutive positive integers, beginning at 1, should be taken to make thesum equal to 2016?

Solution :
1 + 2 + 3 + ...... + n = 2016

T(1) = a = 1
d = 1

S(n):
n[2a + (n - 1)d]/2 = 2016
n[2*1 + (n - 1)*1]/2 = 2016
n[2 + (n - 1)] = 2016*2
n² + n - 4032 = 0
(n - 63)(n + 64) = 0
n = 63 or n = -64 (rejected)

Hence, no. of consecutive positive integers = 63


27.
Theinterior angles of a polygon form an arithmetic sequence. The largest angle is106˚ and the common differenceis 5˚
(a)Find the number of sides of the polygon.
(b)Find the smallest interior angle.

Solution :
T(1) = a
d = 5

T(n) :
a + (n - 1)d = 106
a + (n - 1)*5 = 106
a + 5n - 5 = 106
a = 111 - 5n ...... [1]

S(n):
n[T(1) + T(n)]/2 = (2n - 4)*90
n(a + 106)/2 = 180n - 360
n(a + 106) = 360n - 720 ...... [2]

Put [1] into [2] :
n(111 - 5n + 106) = 360n - 720
n(217 - 5n) = 360n - 720
217n - 5n² = 360n - 720
5n² + 143n - 720 = 0
n = (-143 ± √34849)/10

n must be a positive integer. Hence, there is no solution.

2011-09-18 19:01:53 補充：
In Q3, it should be "160°" instead of "106°" !

(a)
T(1) = a
d = 5

T(n) :
a + (n - 1)*5 = 160
a = 165 - 5n ...... [1]

S(n):
n(a + 160)/2 = 180n - 360
n(a + 160) = 360n - 720 ...... [2]

2011-09-18 19:02:03 補充：
Put [1] into [2] :
n(165 - 5n + 160) = 360n - 720
5n² + 35n - 720 = 0
n² + 7n - 144 = 0
(n + 16)(n - 9) = 0
n = -16 (rejected) or n = 9 ...... (answer)

Q27 has problem