Question

let {Wn}, n is a subset of Z+, be a geometric sequence with the first term p and common ratio q where p > 0 and q > 0. Let another sequence {Zn} be defined by Zn = ln Wn. Find sigma Zi i=1 n giving your answer in the form of ln k with k in terms if n, p and q

Answer 1

Given : {Wn}, n is a subset of Z+, be a geometric sequence with the first term p and common ratio q where p > 0 and q > 0

{Zn} be defined by Zn = ln Wn

To Find : $\sum_{i=1}^nZ_{n}$    in the form of ln k with k in terms if n, p and q

Solution:

W₁ = p

W₂ = pq

W₃ = pq²

Wₙ = Pqⁿ⁻¹

ln(ab) = ln(a) +ln(b)

ln(aⁿ) = nln(a)

Z₁ = ln W₁ = ln(p) = ln(p)

Z₂ = ln W₂ = ln(pq) =  ln(p) +  ln(q)

Z₃ = ln W₃ = ln(pq²) =   ln(p) +  ln(q²)  =  ln(p) + 2 ln(q)

Zₙ = ln Wₙ = ln(pqⁿ⁻¹) =  ln(p) + (n-1) ln(q)

$\sum_{i=1}^nZ_{n}$ = Z₁  + Z₂ + Z₃  + ______ + Zₙ

=  ln(p) + ( ln(p) +  ln(q)) + (ln(p) + 2 ln(q)) + _____ + ( ln(p) + (n-1) ln(q) )

= n  ln(p)  +  ln(q)(1  + 2 +  _____ + (n-1) )

=   ln(pⁿ)  + { (n-1)n/2 } ln(q)

=   $\ln(p^nq^{\frac{n(n-1)}{2}})$

comparing with ln(k)

Hence  $k= p^nq^{\frac{n(n-1)}{2}}$

$\sum_{i=1}^nZ_{n}$  = $\ln(p^nq^{\frac{n(n-1)}{2}})$

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