Question

The numbers 1, 4, 16 can be three terms (not necessarily consecutive) of ? a) no A.P b) only 1 or 2 G.Ps c) infinite number of A.Ps d) infinite number of G.Ps

Answer 1

c) and d)

Explanation:

Arithmetic progressions

For any positive integer `k`, let:

`a_n = 1+(1/k)(n-1)`

Then:

`{ (a_1 = 1), (a_(3k+1) = 4), (a_(15k+1) = 16) :}`

Geometric progressions

For any positive integer `k`, let:

`a_n = 1 * (4^(1/k))^(n-1)`

Then:

`{ (a_1 = 1), (a_(k+1) = 4), (a_(2k+1) = 16) :}`

So c) and d) are both true.