Pre-calc help please!?

Question

Find the sum: 
1 + sqrt3 + 3 +...+ 243

Answer 1

  

The sequence 1 + √3 + 3 + ...+ 243

Common ratio (r) = a₂/a₁  

r = √3/1 = √3

r = 1.732

Formula for Sum of n terms S_n = a(rⁿ - 1)/(r - 1) where r > 1

where a is first term and r is common difference.

 

Now find the value of n.

n^th term in a geometric sequence = ar^n-1

243 = (1)(√3)^n-1

243 = (√3)ⁿ/√3

243(√3) = (√3)ⁿ

(√3)¹⁰(√3) = (√3)ⁿ

(√3)¹¹ = (√3)ⁿ

n = 11

Substitute n = 11, r = √3 and a = 1 in S_n

S₁₁ = (1)[(√3)¹¹ - 1)]/(√3 - 1)

S₁₁ = (243√3 - 1)/(√3 - 1)

= 419.88/0.732

Sum of the sequence 1 + √3 + 3 + ...+ 243 is  S₁₁ = 573.606.