9. Find n so that the nth terms of the following two A.P.’s are the same.
1, 7, 13, 19,g and 100, 95, 90, g.
Answers
AP ( Arithmetic progression).
An arithmetic progression is a list of numbers a1, a2, a3 ………….. an in which each term is obtained by adding a fixed number to the preceding term except the first term.
This fixed number is called the common difference( d ) of the AP. Common difference of an AP will be the difference between any two consecutive terms.
a2= a1+d
a3= a2+d
a4= a3+d
……..
an= an-1+d ………
Each of the numbers in the list is called a term .
Method to find the common difference :
d = a2 - a1 or a3 - a2 or a4 - a3...
General form of an AP.:
a, a+d, a+2d, a+3d…….
Here a is the first term and d is common difference.
General term or nth term of A.P
The general term or nth term of A.P is given by an = a + (n – 1)d, where a is the first term, d is the common difference and n is the number of term.
SOLUTION :
GIVEN - In first sequence
a1 = 1 , a2 = 7 , a3 = 13
(an)or (tn)= a + (n - 1) d
d = t₂-t₁
d = 7-1
d = 6
an or tn = 1 + (n-1) 6
tn = 1 + 6 n - 6
tn = 6 n - 5 ……….. (1)
Given - In second sequence
a1 = 100 , a2 = 95 , a3 = 90
d = t₂-t₁
d = 95 - 100
d = -5
an or tn = 100 + (n-1) (-5)
tn = 100 - 5 n + 5
tn = 105 - 5 n ………. (2)
Given - nth term of two AP’s are same
Eq 1 = eq 2
6 n - 5 = 105 - 5 n
6 n + 5 n = 105 + 5
11 n = 110
n =110/11
n = 10
Hence,11th terms of the given sequence are same.
HOPE THIS WILL HELP YOU….
1 ) 1 , 7 , 13 , 19 , ...is in A.P
First term = a = a1 = 1 ,
Common difference = d = a2 - a1
d = 7 - 1 = 6
nth term = an = a + ( n - 1 )d
an = 1 + ( n - 1 )6
=> an = 1 + 6n - 6
an = 6n - 5 ----( 1 )
ii ) 100 , 95 , 90 , ... is in A.p
a = 100 ,
d = 95 - 100 = -5
nth term = an = 100 + ( n - 1 )( -5 )
=> an = 100 - 5n + 5
an = 105 - 5n ----( 2 )
According to the problem given ,
( 1 ) = ( 2 )
6n - 5 = 105 - 5n
=> 6n + 5n = 105 + 5
=> 11n = 110
n = 110/11
n = 10
Therefore ,
n = 10
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