Question

The 7th term of an AP is 4 times its 2nd term is 2more than 3 times of its 4th term. Find the progression .​

Answer 1

Complete Question:

The 7th term of an AP is four times its 2nd term and 12th term is 2 more than three times of its 4th term. Find the progression.

Answer:

• The progression is: 2, 5, 8, 11, 14, 17 . . . . . . .

Step-by-step explanation:

Given that:

• The 7th term of an AP is four times its 2nd term.
• 12th term is 2 more than three times of its 4th term.

To Find:

• The progression.

Let us assume:

• First term of an AP = a
• Commom difference = d

We know that:

• aₙ = a + (n - 1)d

Where,

• aₙ = nth term
• a = First term
• n = Number of terms
• d = Common difference

According to the question:

The 7th term of an AP is four times its 2nd term.

⟶ a₇ = 4 {a₂}

⟶ a + (7 - 1)d = 4 {a + (2 - 1)d}

⟶ a + 6d = 4 {a + d}

⟶ a + 6d = 4a + 4d

⟶ 6d - 4d = 4a - a

⟶ 2d = 3a _____(i)

12th term is 2 more than three times of its 4th term.

⟶ a₁₂ = 3 {a₄} + 2

⟶ a + (12 - 1)d = 3 {a + (4 - 1)d} + 2

⟶ a + 11d = 3 {a + 3d} + 2

⟶ a + 11d = 3a + 9d + 2

⟶ 11d - 9d = 3a - a + 2

⟶ 2d = 2a + 2

⟶ 3a = 2a + 2 [from eqⁿ (i): 2d = 3a]

⟶ 3a - 2a = 2

⟶ a = 2

∴ First term = 2

In equation (i).

⟶ 2d = 3a

⟶ 2d = 3(2) [The value of a = 2]

⟶ 2d = 6

⟶ d = 6/2

⟶ d = 3

∴ Common difference = 3

Some of the terms of AP:

1. a₁ = a = 2
2. a₂ = a₁ + d = 2 + 3 = 5
3. a₃ = a₂ + d = 5 + 3 = 8
4. a₄ = a₃ + d = 8 + 3 = 11
5. a₅ = a₄ + d = 11 + 3 = 14
6. a₆ = a₅ + d = 14 + 3 = 17

Therefore,

• The progression is: 2, 5, 8, 11, 14, 17 . . . . . . .

Answer 2

Given :-

7th term of AP  is four times its 2nd term

12th term is 2more than 3 times of its 4th term

To find :-

AP

Solution :-

We know that

$\sf a_n = a(n-1)d \;$

$\sf a(7-1) d= 4(a+d)$

$\sf a+6d =4a+4d$

$\sf 4d-6d=a-4a$

$\sf -2d=-3a$

$\sf 2d = 3a$

Now

$\sf a_{12} = 2+3\times a_4$

$\sf a_{12} = 2+3(a+ 3 \times d)$

$\sf a_{12} = 2+3a+9d$

$\sf a+11d = 2+3a+9d$

$\sf  11d-9d=3a-a+2$

$\sf 2d= 2a+2$

Now

3a = 2a + 2

3a - 2a = 2

a = 2

And

d = 6/2

d = 3

Progression = 2,5,8,11