Respuesta :

Answer:

k= 65 and 25

Step-by-step explanation:

(a)

If the sequence is arithmetic then the common difference d between terms is equal

d = 5 - k = k - 125 ( subtract k from both sides )

5 - 2k = - 125 ( subtract 5 from both sides )

- 2k = - 130 ( divide both sides by - 2 )

k = 65

(b)

If the sequence is geometric then the common ratio r between terms is equal

r = [tex]\frac{5}{k}[/tex] = [tex]\frac{k}{125}[/tex] ( cross- multiply )

k² = 625 ( take the square root of both sides )

k = ± [tex]\sqrt{625}[/tex] = ± 25

hence k = 25 ← positive value


gmany

[tex]\text{If}\ a,\ b,\ c\ \text{is an arithmetic progression, then}\ a+c=2b.\\\\\text{If}\ a,\ b,\ c\ \text{is a geometric progression, then}\ ac=b^2.\\\\\text{We have:}\ 125,\ k\ and\ 5.\ Substitute:\\\\(a)\ 2k=125+5\\\\2k=130\qquad\text{divide both sides by 2}\\\\\boxed{k=65}\\\\(b)\ k^2=(125)(5)\\\\k^2=625\to k=\sqrt{625}\\\\\boxed{k=25}[/tex]

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