(a) The first term of an Arithmetic Progression(AP) is 3 and the common difference is 4. Find the sum of the first 28 terms.
(b) Given that (x = frac{2m}{1 – m^{2}}) and (y = frac{2m}{1 + m}), express 2x – y in terms of m in the simplest form.
(c) The angles of pentagon are x°, 2x°, 3x°, 2x° and (3x – 10)°. Find the value of x.
Explanation
(a) (a = 3; d = 4 ; n = 28)
(S_{n} = frac{n}{2}(2a + (n – 1)d))
(S_{28} = frac{28}{2}(2(3) + (28 – 1) 4))
= (14(6 + 108))
= (14(114))
= (1596)
(b) (x = frac{2m}{1 – m^{2}} ; y = frac{2m}{1 + m})
(2x – y = 2(frac{2m}{1 – m^{2}}) – (frac{2m}{1 + m}))
= (frac{4m}{1 – m^{2}} – frac{2m}{1 + m})
= (frac{4m – 2m(1 – m)}{1 – m^{2}})
= (frac{4m – 2m + 2m^{2}}{1 – m^{2}})
= (frac{2m^{2} + 2m}{1 – m^{2}})
= (frac{2m(m + 1)}{(1 + m)(1 – m)})
= (frac{2m}{1 – m})
(c) Sum of interior angles = (2n – 4)90°
When n = 5 (pentagon),
= ((2(5) – 4) times 90°)
= (540°)
(x + 2x + 3x + 2x + (3x – 10) = 11x – 10)
= (11x – 10 = 540°)
= (11x = 550°)
= (x = 50°)