Question

if the 9th term of am A.P be zero then the ratio of its 29th and 19th term is​

Answer 1

Answer:

2:1

hope it helped you....

Answer 2

Given :

The 9th term of an A.P is zero.

To Find :

The ratio of its 29th to 19th term.

Solution :

Analysis :

Here we first we have to form a equation with the help of the information that 9th term of an AP is 0. After forming the equation we can substitute the values that find the required ratio.

Required Formula :

aₙ = a + (n - 1)d

where,

• a = First term
• d = Common Difference
• n = respective term to find
• aₙ = respective term

Explanation :

Let us assume that the first term of the AP be "a" and common difference be "d".

We know that we can find the term by using the formula,

aₙ = a + (n - 1)d

where,

• aₙ = a₉ = 0
• a = a
• d = d
• n = 9

Using the required formula and substituting the required values,

⇒ aₙ = a + (n - 1)d

⇒ a₉ = a + (9 - 1)d

⇒ 0 = a + (8)d

⇒ 0 = a + 8d

⇒ -8d = a

∴ a = -8d ----------- (eq.(i))

Now the 29th term :

aₙ = a + (n - 1)d

where,

• aₙ = a₂₉
• a = -8d
• d = d
• n = 29

Using the required formula and substituting the required values,

⇒ aₙ = a + (n - 1)d

⇒ a₂₉ = a + (29 - 1)d

⇒ a₂₉ = a + (28)d

⇒ a₂₉ = a + 28d ----------- (eq.(ii))

Putting a = -8d in eq.(ii),

⇒ a₂₉ = -8d + 28d

⇒ a₂₉ = 20d

∴ a₂₉ = 20d ----------- (eq.(iii))

Now the 19th term :

aₙ = a + (n - 1)d

where,

• aₙ = a₁₉
• a = -8d
• d = d
• n = 19

Using the required formula and substituting the required values,

⇒ aₙ = a + (n - 1)d

⇒ a₁₉ = a + (19 - 1)d

⇒ a₁₉ = a + 18d ----------- (eq.(iv))

Putting a = -8d in eq.(iv),

⇒ a₁₉ = -8d + 18d

⇒ a₁₉ = 10d

∴ a₁₉ = 10d ----------- (eq.(v))

Now,

From eq.(iii) & eq.(v),

⇒ a₂₉ : a₁₉ = 20d : 10d

⇒ a₂₉/a₁₉ = 20d/10d

Cancelling out the zeroes,

⇒ a₂₉/a₁₉ = 2d/1d

Cancelling out the d,

⇒ a₂₉/a₁₉ = 2/1

⇒ a₂₉ : a₁₉ = 2 : 1

∴ a₂₉ : a₁₉ = 2 : 1.

The ratio of 29th term to 19th term of AP is 2 : 1.