Question

if the 10th term of an hp is 21 and the 21st term of the same hp is 10 then find the 210th term

Answer 1

21=1/(a+9d) and 10=1/(a+20d) cross multiply
21a+189d=1. and. 10a+200d=1. subtract a=d.
substitute a=d=1/210
210 th term =1/(1/210+209/210)=1

Answer 2

Solution:

The 210th terms of the hp is 1.

Explanation:

We know that the nth term of a hp is given by

$a_n=\frac{1}{a+(n-1)d}$

We have been given that  the 10th term of an hp is 21. Thus, we have

$21=\frac{1}{a+(10-1)d}$

$21=\frac{1}{a+9d}$

$a+9d = \frac{1}{21}.................(1)$

Similarly, the 21st term of the same hp is 10. Thus, we have

$10=\frac{1}{a+(21-1)d}$

$10=\frac{1}{a+20d}$

$a+20d = \frac{1}{10}..............(2)$

Subtract equation 1 and 2, we get

$-11d = -\frac{11}{210} \\\\d=\frac{1}{210} \\\\\text{Therefore, the value of a is}\\\\a+20 \times \frac{1}{210} = \frac{1}{10} \\\\a=\frac{1}{10}-\frac{2}{21} \\\\a=\frac{1}{210}$

Therefore, the 210th terms of the hp is given by

$a_{210}= \frac{1}{\frac{1}{210}+(210-1)\frac{1}{210}} \\\\a_{210}=\frac{1}{\frac{210}{210}}\\\\a_{210}=1$