If a1- a2- a3 and a4 are first four terms of an increasing G-P- such that ai- N - i- N and -i-14ai-4a3-a2-32- then find the value of a2-a4a3

Let the common factor be γ . Due to increasing nature of G.P., γ gt;1 .Let a_1=a, a_2=aγ, a_3=aγ^2 and a_4=aγ^3 Also, (a_2+a_4a_3) = ...

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Byju's Answer

Standard XII

Mathematics

Sequence

If a1, a2, ...

Question

If $a_{1}, a_{2}, a_{3}$ and $a_{4}$ are first four terms of an increasing G.P. such that $a i \in N \forall i \in N$ and $4 \sum i = 1 a_{i} = 4 (a_{3} - a_{2}) + 32$ , then find the value of $(a_{2} + \frac{a_{4}}{a_{3}})$

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Solution

Let the common factor be $γ$ . Due to increasing nature of G.P., $γ > 1$ .
Let $a_{1} = a, a_{2} = a γ, a_{3} = a γ^{2}$ and $a_{4} = a γ^{3}$
Also, $(a_{2} + \frac{a_{4}}{a_{3}}) = γ (a + 1)$
It is given $\sum_{i = 1}^{4} a_{i} = 4 (a_{3} - a_{2}) + 32$
$⟹ a \frac{γ^{4} - 1}{γ - 1} = 4 (a γ^{2} - a γ) + 32$
$⟹ a (γ^{3} - 3 γ^{2} + 5 γ - 1) = 32$

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If a1,a2,a3 and a4 are first four terms of an increasing G.P. such that ai∈N∀i∈N and 4∑i=1ai=4(a3−a2)+32, then find the value of (a2+a4a3)

Let the common factor be γ. Due to increasing nature of G.P., γ>1.Let a1=a, a2=aγ, a3=aγ2 and a4=aγ3Also, (a2+a4a3)=γ(a+1)It is given ∑4i=1ai=4(a3−a2)+32⟹aγ4−1γ−1=4(aγ2−aγ)+32⟹a(γ3−3γ2+5γ−1)=32

If $a_{1}, a_{2}, a_{3}$ and $a_{4}$ are first four terms of an increasing G.P. such that $a i \in N \forall i \in N$ and $4 \sum i = 1 a_{i} = 4 (a_{3} - a_{2}) + 32$ , then find the value of $(a_{2} + \frac{a_{4}}{a_{3}})$