Question

The 16th term of an AP is three times the 5th term.If the 12th term is 20 more the 7th term.Find the first term and the common difference​

Answer 1

• First term = 6
• Common difference= 4

Solution:

Let's take the first term as a and the common difference be d

The formula to find the nth term of an arithmetic progression is given by :

• an = a + ( n - 1 ) d

According to question, we have :

• 16th term of an AP is three times the 5th term.

➝ a16 = 3 ( a5 )

➝ a + ( 16 - 1 )d = 3 ( a + ( 5 - 1 )d )

➝ a + 15d = 3 ( a + 4d )

➝ a + 15d = 3a + 12d

➝ 15d - 12d = 3a - a

➝ 3d = 2a

➝ 2a = 3d

➝ a = 3d/2‎ㅤ‎ㅤ‎ㅤ⸻ ( 1 )

• And, the 12th term is 20 more the 7th term.

➝ a12 - ( a7 ) = 20

➝ a + ( n - 1 )d -( a + ( n - 1 )d ) = 20

➝ a + ( 12 - 1 )d - + a + ( 7 - 1 )d ) = 20

➝ a + 11d - ( a + 6d ) = 20

➝ a - a + 11d - 6d = 20

➝ 5d = 20

➝ d = 20/ 5

➝ d = 4

• Using equation ( 1 )

➝ a = 3d / 2

➝ a = ( 3 × 4 ) / 2

➝ a = 12 / 2

➝ a = 6