The 9th term of an A.P. is equal to 6 times its second term. If its 5th term is 22, find the A.P.
Answers
Answer:
The required AP is 2, 7 ,12, 17 , ....
Step-by-step explanation:
Given:
a9 = 6(a2) and a5 = 22
Let the first term of an A.P be 'a' & Common difference be 'd'.
By using the formula , nth term , an = a + (n -1)d
Case : 1
a9 = 6(a2)
a + (9 -1)d = 6 ( a + (2 - 1) d
a + 8d = 6(a + d)
a + 8d = 6a + 6d
a - 6a + 8d - 6d = 0
-5a + 2d = 0 ………..(1)
Case : 2
a5 = 22
a + (5 - 1)d = 22
a + 4d = 22
a = 22 - 4d…………(2)
On putting the value of 'a' in eq 1.
-5a + 2d = 0
-5(22 - 4d) + 2d = 0
- 110 + 20d + 2d = 0
- 110 + 22d = 0
22d = 0 + 110
22d = 120
d = 110/22
d = 5
On putting the value of 'd' in eq 2,
a = 22 - 4d
a = 22 - 4(5)
a = 22 - 20
a = 2
First term , a = 2
Second term ,a2 = a + d = 2 + 5 = 7
Third term , a3 = a2 + d = 7 + 5 = 12
Fourth term, a4 = a3 + d = 12 + 5 = 17
Hence, the required AP is 2, 7 ,12, 17 , ....
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Answer: 2, 7 ,12, 17..
Step-by-step explanation:
Given:
a9 = 6 × (a2) & a5 = 22
a = First term, d = common difference
Using the formula,
an = a + (n -1)d
a9 = 6 × (a2)
» a + (9 - 1)d = 6 (a + (2 - 1)) d
» a + 8d = 6(a + d)
» a + 8d = 6a + 6d
» a - 6a + 8d - 6d = 0
» -5a + 2d = 0..(1)
Also,
a5 = 22
» a + (5 - 1)d = 22
» a + 4d = 22
» a = 22 - 4d..(2)
Putting the value of 'a' in (1),
-5a + 2d = 0
» -5(22 - 4d) + 2d = 0
» - 110 + 20d + 2d = 0
» - 110 + 22d = 0
» 22d = 0 + 110
» 22d = 110
» d = 5
Similarly, Putting the value of 'd' in (2),
a = 22 - 4d
» a = 22 - 4(5)
» a = 22 - 20
» a = 2
d = 5
Thus, we get :
a = 2
a2 = a + d = 2 + 5 = 7
a3 = a2 + d = 7 + 5 = 12
a4 = a3 + d = 12 + 5 = 17
Thus, The required AP is 2, 7, 12, 17..