Find Geometric Sequence from the Given Two Terms :
In this section, we will learn how to find the geometric sequence from the given two terms.
Example 1 :
If the 4^{th} and 7^{th} terms of a G.P are 54 and 1458 respectively, find the G.P
Solution :
4^{th} term = 54
7^{th} term = 1458
t_{4} = 54
a r^{3} = 54
----- (1)
t_{7} = 1458
a r^{6} = 1458 ----- (2)
(2)/(1) = (a r^{6})/(a r^{3}) ==> 1458/54
r^{3} = 27
r = 3
By substituting r = 3 in the first equation we get
a (3)^{3 } = 54
a(27) = 54
a = 54/27
a = 2
The general form of G.P is a, a r , a r ²,.........
= 2, 2(3), 2(3)^{2},..............
= 2,6,18,............
Therefore the required geometric sequence is
2, 6, 18, .......
Example 2 :
In a geometric sequence, the first term is 1/3 and the sixth term is 1/729, find the G.P
Solution :
1^{st }term = 1/3
6^{th} term = 1/729
a = 1/3
t_{6 } = 1/729
a r^{5} = 1/729 ------(1)
Applying the value of a in (1) we get,
(1/3) r^{5} = 1/729
r^{5} = (1/729) / (1/3)
r^{5} = (1/729) ⋅ (3/1)
r^{5} = 1/243
r^{5} = (1/3)^{5}
r = 1/3
The general form of G.P is a, ar, ar^{2},.........
= 1/3, (1/3) (1/3), (1/3)(1/3)²,..............
= 1/3, 1/9, 1/27,............
Example 3 :
The fifth term of a G.P is 1875.If the first term is 3,find the common ratio.
Solution :
Fifth term (t_{5}) = 1875
ar^{4} = 1875 ----(1)
First term (a) = 3
By applying the value of a in (1), we get
3r^{4 }= 1875
r^{4} = 1875/3
r^{4} = 625
r^{4} = 5^{4}
r = 5
Therefore the common ratio is 5.
After having gone through the stuff given above, we hope that the students would have understood how to find the geometric sequence from the given two terms.
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