# The 1st and 2nd term of a geometric sequence are p and sp, in that order. What is the 734th term of the sequence? A (sp)233 B s733P?

Relevance
• T(1)=p, T(2)=sp in a G.P. then the common ratio

r=sp/p=s

=>

T(734)=p(s^733).

• If the first term is "p" and the second term is "sp", then the common ratio is "s".

I use this to describe the n'th term of a geometric sequence:

a(n) = a r^(n - 1)

Where "a" is the first term and "r" is the common ratio.

If we substitute the variables "p" and "s" in place, we get:

a(n) = p s^(n - 1)

The 734th term is then:

a(734) = p s^(734 - 1)

a(734) = p s^733

• For a geometric progression:

a₁ = p

a₂ = s * p → where s is the common ratio

a₃ = a₂ * s = a₁ * s * s = a₁ * s²

a₄ = a₃ * s = a₁ * s² * s = a₁ * s³ → you can generalize by writing:

a(n) = a₁ * s^(n - 1) → recall: a₁ = p

a(n) = p * s^(n - 1) → for the 734th term → n = 734

a₇₃₄ = p * s^(733)

• nth term is ps^(n-1)

• General geometric progression is of form a,ar,ar^2,....,ar^(n-1),...where a is term 1,

ar^(n-1) is the nth term and r = ratio between successive terms {t(n+1)/t(n),n =1,2,3...}. Here, a =p, r = p, nth term =t(n) =ps^(n-1). For n = 744, t(734) = ps^(733).

Your choices A (sp)233 and B s733P are meaningless!!!.

• If term 1 is p and term 2 is sp, the common ratio is s

nth term is psⁿ⁻¹

so, with n = 734 we have:

ps⁷³³....or s⁷³³p

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