Question

Write the first five terms of the sequence given by `(sqrt(3))^n, n ∈ N`.

1. Is the sequence an A.P. or G.P?
2. If the sum of its first ten terms is `p(3 + sqrt(3))`, find the value of p.

Answer 1

Given: Sequence `(sqrt(3))^n, n ∈ N` taken with n = 1, 2, 3,...

Step-wise calculation:

1. First five terms take n = 1 to 5

n = 1: `(sqrt(3))^1 = sqrt(3)`

n = 2: `(sqrt(3))^2 = 3`

n = 3: `(sqrt(3))^3 = 3sqrt(3)`

n = 4: `(sqrt(3))^4 = 9`

n = 5: `(sqrt(3))^5 = 9sqrt(3)`

So, the first five terms are `sqrt(3), 3, 3sqrt(3), 9, 9sqrt(3)`.

2. Determine A.P. or G.P.

Ratio of consecutive terms = `3/sqrt(3) = sqrt(3), (3sqrt(3))/3 = sqrt(3)`, etc. 

The ratio is constant `sqrt(3)`, so the sequence is a geometric progression (G.P.) with first term `a = sqrt(3)` and common ratio `r = sqrt(3)`.

3. Sum of first ten terms and find p if `S_10 = p(3 + sqrt(3))`

For a G.P., `S_n = (a(r^n - 1))/(r - 1)`.

Here, `a = sqrt(3), r = sqrt(3), n = 10`.

Compute `r^10 = (sqrt(3))^10`

= `((sqrt(3))^2)^5`

= 3⁵

= 243

`S_10 = sqrt(3) xx (243 - 1)/(sqrt(3) - 1)`

= `sqrt(3) xx 242/(sqrt(3) - 1)`

Rationalize: multiply numerator and denominator by `(sqrt(3) + 1)`: 

`S_10 = sqrt(3) xx 242 xx (sqrt(3) + 1)/((sqrt(3) - 1)(sqrt(3) + 1))`

= `sqrt(3) xx 242 xx (sqrt(3) + 1)/(3 - 1)`

= `sqrt(3) xx 242 xx (sqrt(3) + 1)/2`

= `121 xx sqrt(3) xx (sqrt(3) + 1)`

= `121 xx (3 + sqrt(3))`

Thus, `S_10 = 121(3 + sqrt(3))`, so p = 121.