Question

In an AP, ${\sf{t_n}}$ represent ${\sf{n^{th}}}$ terms. ${\sf{t_1}}$ and ${\sf{t_3}}$ are two digit numbers such that their digits are same but their position are interchanged.

If ${\sf{t_3 - t_1 = -18}}$ then the sum of the digits of ${\sf{t_1}}$ is 8 then the value of ${\sf{\dfrac{t_2}{t_6}}}$


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Answer 1

EXPLANATION.

In an A.P.

nth term t₁ and t₃ are two number.

Their digit are same but the positions are different.

As we know that,

We can write equations as,

⇒ t₁ = 10x + y.

⇒ t₃ = 10y + x.

⇒ t₃ - t₁ = - 18. [given].

⇒ (10y + x) - (10x + y) = - 18.

⇒ 10y + x - 10x - y = - 18.

⇒ 9y - 9x = - 18.

⇒ y - x = - 2. - - - - - (1).

Sum of the digit of t₁ = 8.

⇒ x + y = 8. - - - - - (2).

From equation (1) & (2), we get.

⇒ y - x = - 2. - - - - - (1).

⇒ x + y = 8. - - - - - (2).

Adding equation (1) & (2), we get.

⇒ 2y = 6.

⇒ y = 3.

Put the value of y = 3 in the equation (2), we get.

⇒ x + y = 8.

⇒ x + 3 = 8.

⇒ x = 8 - 3.

⇒ x = 5.

Values of x = 5 and y = 3.

⇒ t₁ = 10x + y.

⇒ t₁ = 10(5) + 3 = 53.

⇒ t₃ = 10y + x.

⇒ t₃ = 10(3) + 5 = 35.

⇒ t₁ = a = first term = 53.

⇒ t₃ = a + 2d.

⇒ a + 2d = 35.

⇒ 53 + 2d = 35.

⇒ 2d = 35 - 53.

⇒ 2d = - 18.

d = - 9. = common difference.

⇒ t₂/t₆ = (a + d)/(a + 5d).

⇒ t₂/t₆ = (53 - 9)/(53 + 9(-5)) = 35/8 = 4.375 ≈ 4.5.

Option [B] is correct answer.

Answer 2

Answer:

Given Question :

In an AP, ${\sf{t_n}}$ represent ${\sf{n^{th}}}$ terms. ${\sf{t_1}}$ and ${\sf{t_3}}$ are two digit numbers such that their digits are same but their position are interchanged.

If ${\sf{t_3 - t_1 = -18}}$ then the sum of the digits of ${\sf{t_1}}$ is 8 then the value of ${\sf{\dfrac{t_2}{t_6}}}$

Required Answer :

In an A.P

nth term and ${\sf{t_3}}$are two numbers

As we know that ,

↦${\sf{t_1}}$ = 10x + y

↦${\sf{t_3}}$ = 10y + x

Now ,

↦(10y + x) - (10x + y) = -18

↦10y + x - 10x - y = -18

↦9y - 9x = -18

↦y - x = 2 •••••••• equation (1)

Sum of the digit of ${\sf{t_n}}$ is 8

↦x + y = 8 ••••••• equation (2)

Now,

From equation (1) and (2) ,

↦y - x = -2 •••••••equation (1)

↦x + y = 8••••••• equation (2)

Add equation (1) and (2) ,

↦2y = 6

↦y = 3

The value of y = 3 in equation (2) ,

↦y + x = 8

↦3 + x = 8

↦x = 8 - 3

↦x = 5

☘ Value of y = 3 and x = 5

↦${\sf{t_1}}$ = 10x + y

↦${\sf{t_1}}$ = 10(5) + 3 = 53

↦${\sf{t_3}}$ = 10y + x

↦${\sf{t_3}}$ = 10 (3) + 5 = 35

↦${\sf{t_1}}$ = a = first term = 52

↦${\sf{t_3}}$ =a +2d

↦a + 2d = 35

↦53 + 2d = 35

↦35 - 53 = 2d

↦2d = -18

↦$\sf \frac{ t_{2}}{t6} = \frac{a + d}{a + 5d}$

↦$\sf \: \frac{ t_{2}}{t_{6}} = \frac{53 - 9}{53 + 9( - 5)}$

↦$\sf \: \frac{35}{8}$

↦$\sf 4.375≈ \pink{4.5}$

Hence, The answer is 4.5 option [b]