To Find the eth Term From The End Of An A.P. –
Theorem.2) - Let ‘a’ be the 1st term, ‘d’ be the common difference and ‘l’ be the last term of a given A.P. Then, prove that its eth term from the end is {l – (e – 1) d}.
Proof - We may write the given A.P. as
a, (a + d), (a + 2d), (a + 3d),……………(l – 2d), (l – d), l.
then we have
last term = l = [l – (1 – 1) d
2nd term from the end = l – d = [l – (2 – 1) d]
3rd term from the end = l – 2d = [l – (3 – 1) d]
…………… ……………. …………………
…………… ………………. …………………
So, eth term from the end = l – (e – 1) d
Example.) Find the 10th term from the end of the A.P. 4, 9, 14, ………….., 254
Ans.) The given A.P. is 4, 9, 14,………………….
Where, a = 4, d = (9 – 4) = (14 – 9) = 5 (constant), l = 254, and e = 10.
Now, 10th term from the end = {l – (e – 1) d}
= [254 – {(10 – 1) X 5}]
= {254 – (9 X 5)}
= (254 – 45) = 209
Hence, the 10th term from the end is 209. (Ans.)
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