Using a very nice idea in progressions, you can cut down on a lot of working for this question. Remember, Sum up to 'n' terms, is 'n' times the middle term.
2. Progressions
Let the nth term of an A.P. be defined as tn and sum up to n terms be
defined as Sn. If |t8| = |t16| and t3 is not equal to t7, what is S23?
(a) 23 (t16 - t8) (b) 0
(c) 23 times t11 (d) Cannot be determined
3. Progressions
Let the nth term of an A.P. be defined as tn and sum up to n terms be
defined as Sn. If |t8| = |t16| and t3 is not equal to t7, what is S23?
|t8|=|t16|.This can happen under two scenarios t8 = t16 or t8 = – t16.
If t8 = t16, the common difference would be 0 suggesting that t3
would be equal to t7. However, we know that t3 is not equal to t7, so
the common difference cannot be zero.
This tells us that t8 = – t16 Or, t8 + t16 = 0
If t8 + t16 = 0, then t12 = 0.
4. Progressions
Let the nth term of an A.P. be defined as tn and sum up to n terms be
defined as Sn. If |t8| = |t16| and t3 is not equal to t7, what is S23?
t12 = t8 + 4d, and t16 – 4d So, t12 =
t8+t16
2
.
For any two terms in an AP, the mean is the term right in between
them. So, t12 is the arithmetic mean of t8 and t16.
So, t12 = 0.
Now, S23 = 23 × t12.
We know that average of n terms in an A.P. is the middle term. This
implies that sum of n terms in an A.P., is n times the middle term.
So, S23 = 0. Answer choice (b)
5. To learn this and other topics, visit
online.2iim.com
