How do I find the nth term from the end of an AP and the sum of all its terms in SSLC Maths?

Question

Karnataka SSLC, Maths chapter Arithmetic Progressions. I can find the nth term from the start using a + (n-1)d. But when the question asks for the 15th term from the last term, or the sum when only first and last term are given, I get stuck. Is there a direct formula?

GradeHunt_Team · Answer

Yes, there is a direct formula for each, and both come from formulas you already know.

Formulas:
nth term from the start: an = a + (n - 1)d
nth term from the end: l - (n - 1)d, where l is the last term
Sum of n terms: Sn = n/2 [2a + (n - 1)d]
or, when the last term is known: Sn = n/2 (a + l)

Why does l - (n - 1)d work? Read the AP backwards. It becomes a new AP with first term l and common difference -d. Put these into a + (n - 1)d and you get l - (n - 1)d.

Worked example: AP 7, 11, 15, ..., 207
Here a = 7, d = 11 - 7 = 4, l = 207.

(a) Number of terms:
207 = 7 + (n - 1)4
200 = (n - 1)4
n - 1 = 50, so n = 51

(b) 15th term from the end:
= l - (15 - 1)d = 207 - 14 x 4 = 207 - 56 = 151

Check another way: the 15th term from the end is the (51 - 15 + 1) = 37th term from the start.
a37 = 7 + 36 x 4 = 7 + 144 = 151. Same answer.

(c) Sum of all terms:
S51 = 51/2 (7 + 207) = 51/2 x 214 = 51 x 107 = 5457

Common mistakes:
1. Writing l - nd instead of l - (n - 1)d.
2. Using Sn = n/2 (a + l) before finding n. Find n first from the last term.
3. Taking d as (first term - second term). It is always (second term - first term).

The check in part (b) takes one line and catches most slips, so do it whenever time allows.

How do I find the nth term from the end of an AP and the sum of all its terms in SSLC Maths?

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