Question

How many terms of the series 2+2√2+4+.... amount to (30+14√2).​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

7 terms of the series 2+2√2+4+.... amount to (30+14√2)

Given:

• 2+2√2+4+....  amount to (30+14√2).

To Find:

• Number of Terms

Solution:

• (xᵃ)ᵇ = xᵃᵇ
• xᵃ.xᵇ= xᵃ⁺ᵇ

Geometric sequence

• A sequence of numbers in which the ratio between consecutive terms is constant and called the common ratio.
• a , ar , ar² , ... , arⁿ⁻¹
• The nth term of a geometric sequence with the first term a and the common ratio r is given by:   aₙ = arⁿ⁻¹
• Sum is given by  Sₙ = a(rⁿ - 1)/(r - 1)
• Sum of  infinite series is given by  a/(1 - r)   where -1 < r < 1

Step 1:

Find common ration

2√2/2 = √2

4/2√2 = √2

Hence this is an GP with

a = 2

r = √2

Step 2:

Find sum of n terms and equate with 30+14√2 and solve for n

a(rⁿ - 1)/(r - 1)  = Sₙ  

2((√2)ⁿ - 1)/(√2 - 1)  = 30+14√2

(√2)ⁿ - 1  = (15 + 7√2)(√2 - 1)

=> (√2)ⁿ - 1  =  15√2 - 15 + 14 - 7√2

=>  (√2)ⁿ - 1  =  8√2 - 1

=>  (√2)ⁿ  =  8√2  

=> (√2)ⁿ  =  2³√2  

=> (√2)ⁿ  =  ((√2)²)³√2  

=> (√2)ⁿ  =  (√2)⁶√2  

=> (√2)ⁿ  =  (√2)⁷

=> n = 7

Hence, 7 terms of the series 2+2√2+4+.... amount to (30+14√2)