G.P. : = 3 - 6 + 12 - 24 + …………. - 384

Statement (1): Product of 5^th term from the beginning and 5^th term from the end of the G.P. is - 1152.

Statement (2): In an G.P. the product of n^th term from the beginning and n^th term from the end is

1^st term + last term

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Given, G.P.: = 3 - 6 + 12 - 24 + …………. - 384

Here, a = 3

common ratio, r = $\dfrac{-6}{3}$ = -2

a_n = -384

Using the formula; T_n = a.r^{n - 1}

⇒ 3 x (-2)^{n - 1} = -384

⇒ (-2)^{n - 1} = $-\dfrac{384}{3}$

⇒ (-2)^{n - 1} = -128

⇒ (-2)^{n - 1} = (-2)⁷

⇒ n - 1 = 7

⇒ n = 7 + 1 = 8

5_th term from the beginning,

⇒ T₅ = 3 x (-2)^{5 - 1}

= 3 x (-2)⁴

= 3 x 16 = 48

5_th term from the last = (8 - 5 + 1) = 4_th term from the beginning,

⇒ T₄ = 3 x (-2)^{4 - 1}

= 3 x (-2)³

= 3 x (-8) = -24

Product of 5_thterm from the beginning and 5_thterm from the end = 48 x (-24) = -1152.

So, statement 1 is true.

Let in a G.P.

a be the first term and N be the total number of terms.

n^th term from the beginning,

⇒ T_n = a.r^{n - 1} ……..(1)

n^th term from the end,

⇒ T_{N - n + 1} = a.r^{N - n + 1 - 1}

⇒ T_{N - n + 1} = ar^{N - n} …..(2)

Multiplying equation (1) and (2), we get :

⇒ T_n x T_{N - n + 1} = a.r^{n - 1} x a.r^{N - n}

= a².r^{(n - 1) + (N - n)}

= a².r^{N - 1} ………………..(3)

Now, first term + last term = a + a.r^{n - 1}

= a(1 + r^{n - 1}) ………………..(4)

From equation (3) and (4),

The product of n^th term from the beginning and n^th term from the end is not equal to 1^st term + last term.

So, statement 2 is false.

Hence, option 3 is the correct option.