Each term of a certain sequence is 3 less than the previous term. The

Question

Each term of a certain sequence is 3 less than the previous term. The first term of this sequence is 19. If the sum of the first n terms of the sequence is n, what is the value of positive integer n?

A 1
B 13
C 15
D 19
E 47

Bunuel · Accepted Answer

We have the following evenly spaced set (aka arithmetic progression): 19, 16, 13, 10, 7, 4, 1, -2, -5, -8, -11, -14, -17, ... The first term is 19 and the common difference, d, of successive members is -3,

The sum of the elements in any evenly spaced set is given by:  Sum=\frac{a_1+a_n}{2}*n , the mean multiplied by the number of terms.

Thus, we are given that  \frac{a_1+a_n}{2}*n=n  -->  a_1+a_n=2  --> since also given that  a_1=19 , then  a_n=-17 .

The  n_{th}  term of the sequence is given by  a_ n=a_1+d(n-1)  -->  -17=19-3(n-1)  -->  n=13 .

Answer: B.

KarishmaB · Answer

Another method is to ballpark it. 
We know that it is a decreasing sequence i.e. the terms keep decreasing till 0 and then negative terms start.

What does this imply? "If the sum of the first n terms of the sequence is n"
Since number of terms will definitely be positive, we are looking for a positive sum.

19 + 16 + 13 + 10 + 7 + 4 + 1 -2 -5 -7 -10 .... and so on
Note that the first 7 terms are positive and all others negative. Every negative term has greater absolute value than the corresponding positive terms i.e. -2 absolute value is greater than 1 absolute value, -5 absolute value is greater than 4 absolute value, and so on...
Since we have 7 positive terms, we must have less than 7 negative terms to get the sum as positive. If we have 6 negative terms, we will have a total of 13 terms. Of the given options, only 13 is possible and hence it must be the answer.

Answer (B)

PareshGmat · Answer

19 .... 16.... 13 .......... 10 ......... 7.......... 4 .......... 1 ..........  -2 .......... -5 .......... -8 ........... -11 .......... -14............. -17

6 terms to the left of 1

6 terms to the right of 1

Total = 6+6+1 = 13

Corresponding terms addition = 4-2 = 7-5 = 10-8 = 2 .......... so on

Total = 1 + 2*6 = 13

Answer = 13

mvictor · Answer

this one can be solved logically as well..
1 definitely can't be
47 as well..as it would be way to down in the negative side..and thus n can't be negative.

now..
19, 16, 13, 10, 7, 4, 1, -2, -5, -8, -11, -14, -17, -20, -23... list all these numbers..to go further is pointless as the sum would be negative...
19-17=2
16-14=2
13-11=2
10-8=2
7-5=2
4-2=2
1
add these up = 13.
13 numbers, sum is 13. so B it is.

ScottTargetTestPrep · Answer

We have an arithmetic sequence, which has a general formula for the nth term as: a(n) = a(1) + d*(n - 1), where a(1) is the first term and d is the common difference. In this question, a(1) = 19 and d = -3.

We can let a(1) = 19, a(2) = 16, a(3) = 13 and so on. Therefore, the nth term, a(n) = 19 - 3*(n - 1) = 22 - 3n. Since the sequence is evenly spaced, we can use the formula: sum = average x quantity. Since the average of a finite evenly spaced sequence of n terms is   / 2, we have:

n = (  / 2) x n

1 = (41 - 3n) / 2

2 = 41 - 3n

3n = 39

n = 13

Answer: B

whatsinthename1 · Answer

It is asking if the sum of n number is equal to n (statement 1)

Sum = average x n (statement 2), here we know that median = average since the common difference is the same. Common difference being -3

Combine (st. 1) & (st. 2) we get, n = median x n, median =1, therefore we need to find a number whose median is 1

(A) 1 ==> 19, 19 is the median
 (B) 13 ==> to find the median we need the 6th term. We can use the formula a1+n(d), 19 + 6(-3) = 1. Here median is 1 so this is the answer

Each term of a certain sequence is 3 less than the previous term. The

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Each term of a certain sequence is 3 less than the previous term. The

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards