Question

Help solve this plssss

Answer 1

`\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=√(625)\\\\\boxed{k=25}`

Answer 2

k= 65 and 25

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 =
`(5)/(k)` =
`(k)/(125)` ( cross- multiply )

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

k = ±
`√(625)` = ± 25

hence k = 25 ← positive value