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Progressions
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
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.
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)
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Progressions - Working with the Mean

  • 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)
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