Answer:
573.578
Step-by-step explanation:
Geometric Sequence is the sequence in which every digit is the same multiplier of its previous digit.
The given Sequence is: √3 + 3 + 3√3 + ⋯ + 243
here a₁ = √3, r = 3 ÷ √3 = √3.
First we will find the number of terms for this we use formula:
[tex]a_{n} = a_{1}(r)^{n-1}[/tex]
⇒ 243 = √3(√3)ⁿ⁻¹
⇒ (√3)¹⁰ = √3(√3)ⁿ⁻¹
⇒ (√3)⁹ = (√3)ⁿ⁻¹
⇒ n - 1 = 9
⇒ n = 10
The formula of sum of geometric series is:
[tex]S_n=\frac{a_1(1-r^n)}{1-r}[/tex]
⇒ [tex]S_n=\frac{\sqrt{3}(1-\sqrt{3}^{10})}{1-\sqrt{3}}[/tex]
⇒ Sₙ = 572.578147716
Thus the sum of 1 + √3 + 3 + 3√3 + ⋯ + 243 = 1 + 572.578 = 573.578
