Mixed Arithmetic and Geometric sequence

[Einriv]

Can you help me?
Find a sequence of 4 numbers, the first of which is 6 and the fourth is 16. if the first three numbers form an arithmetic progression and the last three numbers form a geometric progression, what is the sequence? show your solution.

 

[Lost souls]

Can you help me?
Find a sequence of 4 numbers, the first of which is 6 and the fourth is 16. if the first three numbers form an arithmetic progression and the last three numbers form a geometric progression, what is the sequence? show your solution.

let the second and third terms be B and C.

since 6,B and C are in AP, hence,

B=6+x, C=6+2x where x is the common difference of the AP.

also B,C and 16 are in GP. .

since the ratios of consecutive terms are equal in a GP,

(6+2x)/(6+x) = 16/(6+2x)

solve for x. .

in case you get a quadratic equation , put the values of x in B and C and check for which value the conditions of AP and GP are satisfied.