the 8th term of a H.P. is 1/31 and its 51st term is 1/289.Find the 21st term of the H.P.
Answers
Step-by-step explanation:
Given :-
The 8th term of a H.P. is 1/31 and its 51st term is 1/289.
To find:-
Find the 21st term of the H.P.?
Solution :-
Given that
8th term of a HP = 1/31
=> 8th term of an AP = 31
Since the terms in the AP are reciprocals to the terms in the HP
We know that
The general term of an AP = an = a+(n-1)d
=> a8 = 31
=> a+(8-1)d = 31
a + 7d = 31---------------(1)
and
51st term of the HP = 1/289
=> 51st term of the AP = 289
=> a 51 = 289
=> a+(51-1)d = 289
a + 50d = 289------------(2)
On subtracting (1) from (2) then
a + 50d = 289
a + 7d = 31
(-)
_____________
0 + 43 d = 258
_____________
=> 43 d = 258
=> d = 258/43
=> d = 6
On Substituting the value of d in (1)
a+ 7(6) = 31
=> a +42 = 31
=> a = 31-42
=> a = -11
We have
a = -11
d = 6
21st term of the AP
=> a 21 = a+20d
=> a 21 = (-11)+20(6)
=> a 21 = (-11)+120
=> a 21 = 120-11
=> a 21 = 109
21st term of the AP = 109
21st term of the HP = 1/109
Answer:-
21st term of the HP for the given problem is 1/109
Used formulae:-
1.The general term of an AP =
an = a+(n-1)d
2.The general form of an AP
= a, a+d,a+2d,..
3.The general form of a HP is 1/a, 1/(a+d),...
- The terms in the AP are reciprocals to the terms in the HP.