Question

Consider the G.P. 3 − 6 + 12 − 24 + ...... − 384. 

Statement (1): The product of the 5^th term from the beginning and the 5^th term from the end is −1152. 

Statement (2): In a G.P., the product of the n^th term from the beginning and the n^th term from the end is the 1^st term + the last term.

विकल्प

• Both the statements are true.

• Both the statements are false.

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

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

Answer 1

Statement 1 is true, and statement 2 is false.

Explanation:

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

Here, a = 3

common ratio, r = `(−6)/3`​ = −2

a_n = −384

Using the formula; T_n = a.r^{n − 1}

⇒ 3 x (−2)^{n − 1} = −384

⇒ (−2)^{n − 1} = `−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 × (−2)^{5 − 1}

= 3 × (−2)⁴

= 3 × 16 = 48

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

⇒ T₄ = 3 x (−2)^{4 − 1}

= 3 × (−2)³

= 3 × (−8) = −24

Product of 5_thterm from the beginning and 5_thterm from the end = 48 × (−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.