In a sequence of numbers t1, t2, t3, ... the difference of any two suc

Question

In a sequence of numbers  t_1 ,  t_2 ,  t_3 , ... the difference of any two successive terms is a constant. If  |t_8| = |t_{16}|  and  t_3 ≠ t_7 , what is the sum of the first 23 terms of the sequence?

A. -10
B. -4
C. 0
D. 4
E. 10

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nick1816 · Accepted Answer

nth term of the sequence =  a+(n-1)d

a= first term 
d= the difference of any two successive terms

t_3 ≠ t_7

a+2d ≠ a+6d

d≠0

|t_8| = |t_{16}|

|a+7d| = |a+15d|

Case 1-  a+7d= a+15d

We get d=0 (not possible)

Reject this case

Case 2-  a+7d= -(a+15d)

2a+22d=0 or  a+11d =0

Sum of first 23 terms = Mean of 23 terms * 23

Mean of 23 terms = 12th term of the sequence = a+11d

Sum of first 23 terms = (a+11d) * 23 =  0

preetamsaha · Answer

In a sequence of numbers t1t1, t2t2, t3t3, ... the difference of any two successive terms is a constant. If |t8|=|t16| and t3≠t7, what is the sum of the first 23 terms of the sequence?

Tn = a + (n - 1) d, where Tn = nth term and a = first term. Here d = common difference = Tn - Tn-1.
Sn =(n/2)

given,|t8|=|t16I
or,|a+7d|=|a+15d|
case I : a+7d= a+15d 
or, d=0  which is not possible because if common difference is zero each term is equal which invalidates t3≠t7 .

so, case II : a+7d= -(a+15d)
or, 2a + 22d = 0

now, the sum of the first 23 terms of the sequence 
=(23/2) 
=23/2 
=0

correct answer C

TestPrepUnlimited · Answer

The scenario here is either  t_8 = -t_{16} , or all terms have a difference of 0. Since  t_3 ≠ t_7  we clearly cannot have a difference of 0, thus we can conclude  t_8 = -t_{16} .

Note there are no actual numbers present in this question,  an educated guess would be to choose C  since we don't have any numbers to conclude an answer like 4 or 10.

In fact we do have  t_9 = -t_{15} ,  t_{10} = -t_{14} , and  t_{11} = -t_{13} , so we must have  t_{12} = 0  . Any sum centered around  t_{12}  would also be equivalent to zero, the terms 1 to 23 have a center of 12 so that sum is indeed equal to 0.

Ans: C

tam87 · Answer

If |t8|=|t16|, then logically one of them is negative and other one is positive, with equal value
In such case, the middle value between t8 and t16, which is t12, must be 0 (this middle value basically indicates the transition from negative to positive or vice versa)

Now, S23 = (23/2) x (2a + 22d) = 23(a + 11d) = 23 x t12 = 0

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sujoykrdatta · Answer

Since the difference between 2 consecutive terms is constant, we can have many possible scenarios, one of which is an Arithmetic progression. 
For example, one series can be: a, b, a, b, a, b ... where the terms alternate. Since we are discussing the difference, we have to basically subtract the smaller from the larger. Thus, the series 1, 4, 1, 4, ... has difference between successive terms always 3. 
In such a scenario, the terms t1 = t3 = t5 = t7 = t9 etc. and t2 = t4 = t6 = t8 ... = t16 
Note that we cannot have this particular case as possible since t3 and t7 are not equal (mentioned). 
However, there can be other variations too; for example: 1, 4, 7, 10, 7, 4, 1, 4, 7 ... also has the difference between consecutive terms equal. 
In such cases, it won't be possible to determine the answer since there can be many possibilities.

Most likely, this question should be interpreted as an Arithmetic sequence.

In that case, we have: |t8| = |t16|=> t8 and t16 are equal but opposite in sign (since they all cannot be equal as it is mentioned that t3 is not equal to t7). 
  Thus, the middle number (between t8 and t16), i.e. t12 must be zero

The average of the first 23 terms is the (23 + 1)/2 = 12th term 
  Thus, the average of the first 23 terms is 0  
=> Sum of the first 23 terms is also 0

Answer C

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