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answers:
la console say: For a geometric sequence:

a₁

a₂ = a₁ * r → where r is the common ratio

a₃ = a₂ * r = a₁ * r * r = a₁ * r²

a₄ = a₃ * r = a₁ * r² * r = a₁ * r³

…and you can generalize writing: a(n) = a₁ * r^(n - 1)


a₃ = a₁ * r² → given that: a₃ = 96

a₁ * r² = 96

a₁ = 96/r² → given that: r = 2

a₁ = 96/4

a₁ = 24



Recall:

a(n) = a₁ * r^(n - 1) → where: n = 7

a₇ = a₁ * r⁶ → we've seen that: a₁ = 24

a₇ = 24 * r⁶ → you know that: r = 2

a₇ = 24 * 2⁶

a₇ = 1536
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Iggy Rocko say: a7 = a3 * r^(7 - 3)
a7 = 96 * 2^4
a7 = 1536
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Puzzling say: Given that the seventh term is close to the third term, you can just double it a few times.
r = 2

a[3] = 96
a[4] = 2 * 96 = 192
a[5] = 2 * 192 = 384
a[6] = 2 * 384 = 768
a[7] = 2 * 768 = 1536

Or you could realize you have to make *4* steps going from a[3] to a[7] so you need to multiply by 2^4 = 16
a[7] = 16 * 96 = 1536

Alternatively, you could use the explicit definition of the nth term and go from there, but in this case, I think that takes longer.
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Dasha say: @Captain Matticus, LandPiratesInc Thank you!(:
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Captain Matticus, LandPiratesInc say: a[3] = 96
a[3] = a[1] * r^2
a[1] * 2^2 = 96
a[1] * 4 = 96
a[1] = 24

a[7] =>
a[1] * r^6 =>
24 * 2^6 =>
3 * 2^3 * 2^6 =>
3 * 2^9 =>
3 * 512 =>
1536
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