Question

The 2nd, 6th, 8th terms of an A.P. form a G.P. , find the common ratio and the general term of the G.P.​

Answer 1

The terms of an arithmetic progression, can form consecutive terms of a geometric progression.

• The common ratio is: $\mathbf{r = \frac{a + 5d}{a + d}}$
• The general term of the GP is: $\mathbf{a_n = (a + d) \times (\frac{a + 5d}{a + d})^{n-1}}$

The nth term of an AP is:

$\mathbf{T_n = a + (n - 1)d}$

So, the 2nd, 6th and 8th terms of the AP are:

$\mathbf{T_2 = a + d}$

$\mathbf{T_6 = a + 5d}$

$\mathbf{T_8 = a + 7d}$

The first, second and third terms of the GP would be:

$\mathbf{a_1 = a + d}$

$\mathbf{a_2 = a + 5d}$

$\mathbf{a_3 = a + 7d}$

The common ratio (r) is calculated as:

$\mathbf{r = \frac{a_2}{a_1}}$

This gives

$\mathbf{r = \frac{a + 5d}{a + d}}$

The nth term of a GP is calculated using:

$\mathbf{a_n = a_1r^{n-1}}$

So, we have:

$\mathbf{a_n = (a + d) \times (\frac{a + 5d}{a + d})^{n-1}}$

Read more about arithmetic and geometric progressions at:

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