Find the sum

Question

a. Find the sum 4,(4/3),(4/9),(4/27),(4/81)

 

b. Find the sum of the first 70 terms

 

 

Please explain both answers in detail

Answer 1

Hi Ethan, as for explanation this is a geometric series with terms (leaving 4 aside as a common factor of course)

 

s₁ = 1/3                                                                            //1

s₂ = (1/3)(1/3)                                                                  //2

s₃ = (1/3)(1/3)(1/3)                                                           //3

s₄ = (1/3)(1/3)(1/3)(1/3)                                                    //4

s₅ = (1/3)(1/3)(1/3)(1/3)(1/3)                                             //5

.                                                                                       .

.                                                                                       .

.                                                                                       .

s₁₀ = (1/3)(1/3)(1/3)(1/3)(1/3)(1/3)(1/3)(1/3)(1/3)(1/3)      //10

Answer 2

S_n = a₁ [ (1 - rⁿ) / (1 - r) ], where a₁ is the first term, r = common ratio, n = number of terms.

 

S₅ = 4 [ ((1 - (1/3)⁵) / (1 - (1/3))]

S₅ = 4 [ (1 - 1/243) / (2/3) ]

S₅ = 4 [ (242/243) / (2/3)

S₅ = 4 [ 121/81 ]

S₅ = 484/81 

 

Sum of the first five terms is 484/81.

 

Complete the second question by substituting 70 for 5.