Three geometric progression's p, q, and r, are such that there sums to infinity are the first three terms respectively of an arithmetic progression.
progression p is 2,1, 1/2,1/
progression q is 3,1,1/3,1/

a. find the sun to infinity of progression r.
b. given that the first term of r is 4, find the sun of the first three terms of r.

S= a(1-r ⁿ) /(1-r) & when r is < 1 & n →∞, the sum becomes S= a/(1-r)

Progression P is 2,1, 1/2,1/4,.. ==> r = 1/2
Sum of P when  n--> ∞ = 2/(1-1/2) ==> S =4

Progression Q is 3,1,1/3,1/9,==> r = 1/3
Sum of Q when  n--> ∞ = 3/(1-1/3) ==> S =4.5

GIVEN THAT 4, 9/2 & X (to be calculated later is a geometric Progression, hence 9/2 - 4 = 0.5 =d (common difference , so X = 4.5+0.5 & X = 5

Sum of R =5 Then => 5=  4 /(1-r)  & r=1/5
Then the sum of R = 4/(1-1/5) ==> S of R =5

As you know john locke said he wanted life liberty and the pursuit of happiness so thomas jefferson was inspired about what john locke said so thomas jefferson wrote the declaration of independence for people to be free for people to be happy

40%=2/550=1/230=3/1056=14/25