e-th Term From The End of G.P. –
Theorem.2) In a given G.P. let the first term be ‘a’, common ratio be ‘r’, and the last term be ‘l’.
l
Then its e-th term from the end is given by -------
r⁽ᵉˉ¹⁾
Proof.- In a given G.P. let the first term be ‘a’, common ratio be ‘r’, and the last term be ‘l’
Then,
l l
2nd term from the end = ------ = -------
r r⁽²ˉ¹⁾
l l
3rd term from the end = ------- = -------
r² r⁽³ˉ¹⁾
l l
4th term from the end = ------- = -------
r³ r⁽⁴ˉ¹⁾
l l
5th term from the end = ------- = -------
r⁴ r⁽⁵ˉ¹⁾
………….… …..………… ……………….
……………. ……………... ……………….
l
e-th term from the end = ------
r⁽ᵉˉ¹⁾
l
Hence the e-th term from the end is ------
r⁽ᵉˉ¹⁾
Example.1) Find the 10th term from the end of the G.P. 3, 6, 12, 24,……….,6144
Ans.) Given G.P. is 3, 6, 12, 24,…………………., 6144
6
Here, a = 3, r = ----- = 2, l = 6144
3
l 6144
10th term from the end = ------ = ---------
r⁽ᵉˉ¹⁾ 2⁽¹⁰ˉ¹⁾
6144
= ----------
2⁹
6144
= ---------
512
= 12
Hence the 10th term from the end is 12 (Ans.)
Your second block of text...