maths questions(for AP and GP)

1)
Let a and d (d > 0) be the first term and the common difference of the arithmetic sequence.
Then the five interior angles are ao, (a+d)o, (a+2d)o, (a+3d)o and (a+4d)o respectively.

Sum of the interior angles = (2x5 - 4) x 90o = 540o

n[2a + (n-1)d]/2 = 540
5[2a + (5-1)d]/2 = 540
5[2a + 4d]/2 = 540
2a + 4d = 216
a + 2d = 108 ...... (1)
Assuming d > 0, a + 3d = 178 ooro a + 4d = 178

If a + 3d = 178 ...... (2)
(2)-(1): d = 70
(1): a + 2(70) = 108
a = -32 (rejected)

If a + 4d = 178 ...... (3)
(3)-(1): 2d = 70
d = 35

(1): a + 2(35) = 108
a = 38
ao = 38o
(a + d)o = 73o
(a + 2d)o = 108o
(a + 3d)o = 143o
(a + 4d)o = 178o

Size of the smallest interior angle = 38o

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2)a)
Let a and d be the first term and the common difference of the arithmetic sequence.

Sum of the first n terms = n[2a + (n-1)d]/2

Sum of the first 4 terms:
4[2a + (4-1)d]/2 = -313
2a + 3d = -156.5 ...... (1)

Sum of the first 20 terms:
20[2a + (20-1)d]/2 = -1165
a + 19d = -116.5 ..... (2)

(2)-(1):
16d = 40
d = 2.5

(1):
2a + 3(2.5) = -156.5
2a = -164
a = -82

a + (n - 1)d
= -82 + (n - 1)(2.5)
= -82 + 2.5n - 2.5
= -84.5 + 2.5n

Ans: The general term of the AP is -84.5 + 2.5n

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2)b)
When the term is positive:
-84.5 + 2.5n > 0
2.5 n > 84.5
n > 33.8

The 34th term is the smallest positive term.

The 34th term
= -84.5 + 2.5(34)
= 0.5

Ans: The smallest positive term of the AP is 0.5 (the 34th term).
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