In a sequence a1,a2,a3,â€¦

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BTGmoderatorLU: Moderator; Posts: 2554; Joined: Sun Oct 15, 2017 1:50 pmFollowed by: 6 members

by BTGmoderatorLU » Sun Oct 15, 2017 3:11 pm

In a sequence a1,a2,a3,..., each term after the first is found by taking the negative of the preceding term, and adding 1. If a1 = 2, what is the sum of the first 99 terms?

(A) 49
(B) 50
(C) 51
(D) 99
(E) 101

The OA is C.

I don't have very clear this PS question. Experts Can you help me to solve it please?

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Source: Beat The GMAT — Problem Solving | Sun Oct 15, 2017 3:11 pm

DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 amFollowed by: 128 members; Location: Boston, MA; Thanked: 1153 times; GMAT Score:770

by DavidG@VeritasPrep » Mon Oct 16, 2017 7:52 am

LUANDATO wrote:In a sequence a1,a2,a3,..., each term after the first is found by taking the negative of the preceding term, and adding 1. If a1 = 2, what is the sum of the first 99 terms?

(A) 49
(B) 50
(C) 51
(D) 99
(E) 101

The OA is C.

I don't have very clear this PS question. Experts Can you help me to solve it please?

First find the pattern:
A1 = 2
A2 = -2 + 1 = -1
A3 = -(-1) + 1 = 2
A4 = -1

So our sequence looks like this: 2, -1, 2, -1, 2...
All the ODD terms = 2. # of ODD terms from A1 - A99 = 50. So 50 terms equal to 2 will give us 2*50 = 100.
All the EVEN terms = -1. # EVEN terms from A2 - A98 = 49. 50 terms equal to -1 = -49.
Total Sum = 100 + (-49) = 51.

The answer is C

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EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pmFollowed by: 12 members; Thanked: 180 times

by EconomistGMATTutor » Mon Oct 16, 2017 1:25 pm

In a sequence a1,a2,a3,..., each term after the first is found by taking the negative of the preceding term, and adding 1. If a1 = 2, what is the sum of the first 99 terms?

(A) 49
(B) 50
(C) 51
(D) 99
(E) 101

The OA is C.

I don't have very clear this PS question. Experts Can you help me to solve it please?

Hi LUANDATO,
Let's take a look at your question.

$$a_1=2$$
$$a_2=-2+1=-1$$
$$a_3=-\left(-1\right)+1=1+1=2$$
$$a_4=-2+1=-1$$
$$a_5=-\left(-1\right)+1=1+1=2$$

Since, all odd terms are 2 and all even terms are -1, therefore,
$$a_{99}=2$$

The sequence will be like,
$$2,\ -1,\ 2,\ -1,\ 2,\ -1,\ ...,\ 2$$
The sequence has 99 terms in total.
It has 49 pairs of two terms 2 and -1 and then a last term 2.
We can write the sum as,
$$Sum = \left(2-1\right)+\left(2-1\right)+\left(2-1\right)+....+\left(2-1\right)+2$$
$$Sum=\left(1\right)+\left(1\right)+\left(1\right)+....+\left(1\right)+2$$
Since, there are 49 ones in the sum above, therefore,
$$Sum=49+2$$
$$Sum=51$$

Therefore, Option C is correct.

Hope this helps.
I am available if you'd like any follow up.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8154; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Fri Nov 22, 2019 11:25 am

BTGmoderatorLU wrote:In a sequence a1,a2,a3,..., each term after the first is found by taking the negative of the preceding term, and adding 1. If a1 = 2, what is the sum of the first 99 terms?

(A) 49
(B) 50
(C) 51
(D) 99
(E) 101

The OA is C.

I don't have very clear this PS question. Experts Can you help me to solve it please?

a1 = 2

a2 = -1

a3 = 2

a4 = -1

a5 = 2

a6 = -1

We see that each odd/even pair sums to positive 1.

There are 49 even/odd pairs in the first 98 terms; hence, the sum is 49. Since a99 is an odd-numbered term, a99 = 2. So the sum of the first 99 terms is 49 + 2 = 51.

Answer: C

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