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For a sequence , if tn= 5^n-2/7^n-3, verify whether
the sequence is a G.P. If it is a G.P., find its
first term and the common ratio.
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Given : a sequence , if tn= 5^n-2/7^n-3,
To Find : first term and the common ratio.
Solution:
Tₙ = 5ⁿ⁻²/7ⁿ⁻³
T₁ = 5⁻¹/7⁻² = 49/5
T₂ = 5⁰/7⁻¹ = 7
T₃ = 5¹/7⁰ = 5
T₄ = 5²/7¹ = 25/7
This is an GP
Tₙ = 5ⁿ⁻²/7ⁿ⁻³
Tₙ₊₁ = 5ⁿ⁺¹⁻²/7ⁿ⁺¹⁻³
=> Tₙ₊₁ = 5ⁿ⁻¹/7ⁿ⁻²
Common Ratio = Tₙ₊₁ /Tₙ
= (5ⁿ⁻¹/7ⁿ⁻²) /( 5ⁿ⁻²/7ⁿ⁻³)
= 5ⁿ⁻¹⁻⁽ⁿ⁻²⁾/7ⁿ⁻²⁻⁽ⁿ⁻³⁾
= 5/7
Common Ratio = 5/7
49/5 is 1st term
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