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प्रश्न
For a sequence Sn = 4(7n – 1), verify whether the sequence is a G.P.
उत्तर
Sn = 4(7n – 1)
∴ Sn–1 = 4(7n–1 – 1)
But, tn = Sn – Sn–1
= 4(7n – 1) – 4(7n–1 – 1)
= 4(7n – 1– 7n– 1 + 1)
= 4(7n–1+1 – 7n–1)
= 4.7n–1 (7 – 1)
∴ tn = 24.7n–1
∴ tn+1 = 24(7)n+1–1
= 24(7)n
The sequence (tn) is a G.P., if `("t"_("n" + 1))/"t"_"n"` = constant for all n ∈ N.
∴ `("t"_("n" + 1))/"t"_"n" = (24(7)^"n")/(24(7)^("n" - 1)`
= 7 = constant, for all n ∈ N
∴ the sequence is a G.P.
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