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Cartesian Product

Find the G.P....

Question

Find the G.P. if

(a) T₃ : T₆ = 1 : 8 and T₅ = 64

(b) T₄ : T8 = 1 : 81 and T₆ = 486

(c) S₁₀ : S₅ = 33 : 1 and T₆= 32

(d) S₈ : S₄ = 97 : 81 and T₅ = 16

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Solution

(a) We need to find the G.P., where and .

Hence, the G.P is 4, 8, 16, 32 …

(b) We need to find the G.P., where and .

Hence, the G.P is 2, 6, 18, 54 …

(c) We need to find the G.P., where and .

However, in a G.P.,

Hence, the G.P is 1, 2, 4, 8 …

(d) We need to find the G.P., where and .

However, in a G.P., r ≠ 1.

Hence, the G.P is 81, 54, 36, 24 …

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Q. Determine two

G . P . s

which satisfy the following conditions:

T_{1} + T_{2} + T_{3} + T_{4} = 30

T_{5} + T_{6} + T_{7} + T_{8} = 480

Also prove the following for both:
(i)

S_{6} - S_{4} = 96

\frac{T_{n}}{T_{n}^{'}} = \frac{1}{3} {(- 1)}^{n}

T_{2} + T_{5} = 216

T_{4} : T_{6} = 1 : 4

and all terms are integers, then its first term is:

T_{2} + T_{5} = 216

T_{4} : T_{6} = 1 : 4

and all the terms are integers, then the first term is

A P, t_{6} = 2 (t_{3}) + 1

t_{4} + t_{5} = 5 (t_{2}) .

9^{t h}

term of the AP is:

In a G.P., $T_{2}$ + $T_{5}$ = 216 and $T_{4}$ : $T_{6}$ = 1:4 and all terms are integers, then its first term is:

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