Home » Mathematics Theory (a) What is the 25th term of 5, 9, 13,… ? (b) Find the 5th…

Mathematics Theory (a) What is the 25th term of 5, 9, 13,… ? (b) Find the 5th…

(a) What is the 25th term of 5, 9, 13,… ?

(b) Find the 5th term of (frac{8}{9}, frac{-4}{3}, 2, …).

(c) The 3rd and 6th terms of a G.P are (48) and (14frac{2}{9}) respectively. Write down the first four terms of the G.P.

Explanation

(a) 5, 9, 13, … is an A.P with the first term = 5 and common difference = 9 – 5 = 13 – 9 = 4.

(T_{n} = a + (n – 1)d) (terms of an A.P)

(T_{25} = 5 + (25 – 1) times 4)

= (5 + 24 times 4)

= ( 5 + 96 = 101)

(b) (frac{8}{9}, frac{-4}{3}, 2, …) is a G.P with first term = (frac{8}{9})

(r = frac{T_{2}}{T_{1}} = -frac{4}{3} div frac{8}{9} = frac{-4}{3} times frac{9}{8} = -frac{3}{2})

(T_{n} = ar^{n – 1}) (terms of a G.P)

(T_{5} = (frac{8}{9})(-frac{3}{2})^{5 – 1})

= (frac{8}{9} times frac{81}{16} = frac{9}{2})

(c) (T_{n} = ar^{n – 1}) (terms of a G.P)

(T_{3} = ar^{2} = 48 … (1))

(T_{6} = ar^{5} = 14frac{2}{9} … (2))

((2) div (1) : r^{3} = frac{128}{9} div 48 = frac{128}{9} times frac{1}{48})

(r^{3} = frac{8}{27})

(r = sqrt[3]{frac{8}{27}} = frac{2}{3})

From (1), (ar^{2} = 48 implies a times (frac{2}{3})^{2} = 48)

(4a = 48 times 9 implies a = frac{48 times 9}{4} = 108)

(T_{2} = 108 times frac{2}{3} = 72)

(T_{4} = 48 times frac{2}{3} = 32)

(therefore) The first four terms of the sequence are (108, 72, 48, 32).