Answer :
The first term of the geometric sequence is 3 and the common ratio is 3/2
Geometric series are is defined as the sum of the sequence of numbers.
The sum of the nth term of a geometric sequence is expressed as:
[tex]S_n=\frac{a(r^n-1)}{r-1}[/tex]
If the sum of the first three terms is 14.25, then;
n = 3
S₃ = 14.25
Substitute into the formula
[tex]S_3=\frac{a(r^3-1)}{r-1}\\14.25=\frac{a(r^3-1)}{r-1} ................. 1[/tex]
Also, if the sum of the first four terms is 24.375, then:
[tex]S_4=\frac{a(r^4-1)}{r-1}\\24.375=\frac{a(r^4-1)}{r-1} ................. 2[/tex]
Divide equations 1 and 2
[tex]\frac{14.25}{24.375}=\frac{r^3-1}{r^4-1} \\\\14.25r^4-14.25=24.3745r^3-24.375\\14.25r^4-24.3745r^3 =-24.375 + 14.25\\ 14.25r^4-24.3745r^3 =-10.125\\ 14.25r^4-24.3745r^3+10.125=0[/tex]
On factorizing:
r = 3/2
substitute r - 3/2 into equation 1
From 1:
[tex]14.25=\frac{a((3/2)^3-1)}{3/2-1} \\14.25=\frac{a(27/8-1)}{3/2-1} \\14.25=\frac{a(19/8)}{0.5} \\\frac{19a}{8} = 14.25 * 0.5\\ \frac{19a}{8} = 7.125\\19a=57\\a=\frac{57}{19}\\a= 3\\[/tex]
This shows that the first term is 3 and the common ratio is 3/2
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