6/2 (2a+5d) = 21
a+6d = 3(a+2d + a+3d)
so you have
a = -9
d = 5
The sum of the first 6 term of AP is 21if the 7th term is 3 times the sum of the 3rd and 4th term .Find the first term and common difference.
2 answers
In an Arithmetic Progression:
an = a + ( n - 1 ) d
where
a = a1 = the initial term
an = the nth term
d = the common difference of successive members
The sum of the first n term:
Sn = n / 2 [ 2 a + ( n - 1 ) d ]
Ginen conditions:
S6 = n / 2 [ 2 a + ( n - 1 ) d ] = 21
6 / 2 [ 2 a + ( 6 - 1 ) d ] = 21
3 ( 2 a + 5 d ) = 21
Divide both sides by 3
2 a + 5 d = 7
The 7th term is 3 times the sum of the 3rd and 4th term means:
a7 = 3 ( a3 + a4 )
Since:
a3 = a + 2 d , a4 = a + 3 d , a7 = a + 6 d
a7 = 3 ( a3 + a4 )
a + 6 d = 3 ( a + 2 d + a + 3 d )
a + 6 d = 3 ( 2 a + 5 d )
a + 6 d = 6 a + 15 d
Subract 15 d d to both sides
a - 9 d = 6 a
Subract a to both sides
- 9 d = 5 a
5 a = - 9 d
Now you must solve system of two equations:
2 a + 5 d = 7
5 a = - 9 d
Try that.
The solution is:
a = - 9 , d = 5
Check result:
a1 = - 9
a2 = - 9 + 5 = - 4
a3 = - 4 + 5 = 1
a4 = 1 + 5 = 6
a5 = 6 + 5 = 11
a6 = 11 + 5 = 16
a7 = 16 + 5 = 21
The sum of the first 6 term:
- 9 + ( - 4 ) + 1 + 6 +11 +16 = - 9 - 4 + 1 + 6 +11 +16 = 21
Correct.
a7 = 3 ( a3 + a4 )
21 = 3 ( 1 + 6 )
21 = 3 ∙ 7
Correct.
an = a + ( n - 1 ) d
where
a = a1 = the initial term
an = the nth term
d = the common difference of successive members
The sum of the first n term:
Sn = n / 2 [ 2 a + ( n - 1 ) d ]
Ginen conditions:
S6 = n / 2 [ 2 a + ( n - 1 ) d ] = 21
6 / 2 [ 2 a + ( 6 - 1 ) d ] = 21
3 ( 2 a + 5 d ) = 21
Divide both sides by 3
2 a + 5 d = 7
The 7th term is 3 times the sum of the 3rd and 4th term means:
a7 = 3 ( a3 + a4 )
Since:
a3 = a + 2 d , a4 = a + 3 d , a7 = a + 6 d
a7 = 3 ( a3 + a4 )
a + 6 d = 3 ( a + 2 d + a + 3 d )
a + 6 d = 3 ( 2 a + 5 d )
a + 6 d = 6 a + 15 d
Subract 15 d d to both sides
a - 9 d = 6 a
Subract a to both sides
- 9 d = 5 a
5 a = - 9 d
Now you must solve system of two equations:
2 a + 5 d = 7
5 a = - 9 d
Try that.
The solution is:
a = - 9 , d = 5
Check result:
a1 = - 9
a2 = - 9 + 5 = - 4
a3 = - 4 + 5 = 1
a4 = 1 + 5 = 6
a5 = 6 + 5 = 11
a6 = 11 + 5 = 16
a7 = 16 + 5 = 21
The sum of the first 6 term:
- 9 + ( - 4 ) + 1 + 6 +11 +16 = - 9 - 4 + 1 + 6 +11 +16 = 21
Correct.
a7 = 3 ( a3 + a4 )
21 = 3 ( 1 + 6 )
21 = 3 ∙ 7
Correct.