 It is currently 18 Sep 2020, 05:13 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. # The sequence A is defined by An = An – 1 + 2 for each intege  Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS: Retired Moderator Joined: 07 Jun 2014
Posts: 4803
GRE 1: Q167 V156 WE: Business Development (Energy and Utilities)
Followers: 171

Kudos [?]: 2911  , given: 394

The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
1
KUDOS
Expert's post 00:00

Question Stats: 55% (00:43) correct 44% (01:48) wrong based on 38 sessions
The sequence A is defined by $$A_n = A_{n – 1} + 2$$ for each integer n ≥ 2, and $$A_1 = 45$$. What is the sum of the first 100 terms in sequence A?

(A) 243
(B) 14,400
(C) 14,500
(D) 24,300
(E) 24,545
[Reveal] Spoiler: OA

_________________

Sandy
If you found this post useful, please let me know by pressing the Kudos Button

Try our free Online GRE Test Retired Moderator Joined: 07 Jun 2014
Posts: 4803
GRE 1: Q167 V156 WE: Business Development (Energy and Utilities)
Followers: 171

Kudos [?]: 2911  , given: 394

Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
2
KUDOS
Expert's post
Explanation

The first term of the sequence is 45, and each subsequent term is determined by adding 2. The problem asks for the sum of the first 100 terms, which cannot be calculated directly in the given time frame; instead, find the pattern.

The first few terms of the sequence are 45, 47, 49, 51,… What’s the pattern? To get to the 2nd term, start with 45 and add 2 once. To get to the 3rd term, start with 45 and add 2 twice. To get to the 100th term, then, start with 45 and add 2 ninety-nine times:

$$45 + (2)(99) = 243.$$

Next, find the sum of all odd integers from 45 to 243, inclusive. To sum up any evenly spaced set, multiply the average (arithmetic mean) by the number of elements in the set. To get the average, average the first and last terms. Since $$\frac{45+243}{2}= 144$$, the average is 144.

To find the total number of elements in the set, subtract 243 – 45 = 198, then divide by 2 (count only the odd numbers, not the even ones): $$\frac{198}{2}= 99$$ terms.

Now, add 1 (to count both endpoints in a consecutive set, first subtract and then “add 1 before you’re done”). The list has 100 terms. Multiply the average and the number of terms:

$$144 \times 100 = 14,400$$
_________________

Sandy
If you found this post useful, please let me know by pressing the Kudos Button

Try our free Online GRE Test Intern Joined: 03 Jan 2019
Posts: 3
Followers: 0

Kudos [?]: 4  , given: 1

Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
2
KUDOS
Alternative Short cut approach:

The first item is 45, which is an odd number, each successive item is plus 2, implying that all 100 terms would be odd numbers. As they are evenly spaced and a total of 100, the mean should be the number between the 50th term and the 51st term. It has to be an even number since it would lie between two Odd numbers. Dividing each of the options should reveal their corresponding mean values. Our answer should be the sum that yields an even mean value. There is only one such option. Supreme Moderator
Joined: 01 Nov 2017
Posts: 371
Followers: 10

Kudos [?]: 175  , given: 4

Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
2
KUDOS
Expert's post
sandy wrote:
The sequence A is defined by $$A_n = A_{n – 1} + 2$$ for each integer n ≥ 2, and $$A_1 = 45$$. What is the sum of the first 100 terms in sequence A?

(A) 243
(B) 14,400
(C) 14,500
(D) 24,300
(E) 24,545

Another method..

This is an AP as terms are evenly spaced..
$$A_1=45$$, so $$A_{100}=45+2(100-1)=45+198=243$$
the average of the sequence is $$\frac{1^{st} term+ 2^{nd}}{2}=\frac{45+243}{2}=\frac{288}{2}=144$$
SUM = average * number of terms = 144*100 = 14,400

B
_________________

Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html Intern Joined: 26 Dec 2018
Posts: 2
Followers: 0

Kudos [?]: 4  , given: 0

Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
3
KUDOS
sandy wrote:
The sequence A is defined by $$A_n = A_{n – 1} + 2$$ for each integer n ≥ 2, and $$A_1 = 45$$. What is the sum of the first 100 terms in sequence A?

(A) 243
(B) 14,400
(C) 14,500
(D) 24,300
(E) 24,545

Solution:

Given A_1 = 45
n ≥ 2,
A_2 = A_1 + 2 = 45+2 = 47

Series {45, 47, 49.....)
First term a = 45 and common difference d = 2
Number of terms = 100

Sum = (n/2)*(2a + (n-1)d)
= (100/2) * (2 x 45 + 99 x 2)
= (100/2) * 288
= 14400
Director  Joined: 22 Jun 2019
Posts: 517
Followers: 4

Kudos [?]: 103 , given: 161

Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
chetan2u wrote:
sandy wrote:
The sequence A is defined by $$A_n = A_{n – 1} + 2$$ for each integer n ≥ 2, and $$A_1 = 45$$. What is the sum of the first 100 terms in sequence A?

(A) 243
(B) 14,400
(C) 14,500
(D) 24,300
(E) 24,545

Another method..

This is an AP as terms are evenly spaced..
$$A_1=45$$, so $$A_{100}=45+2(100-1)=45+198=243$$
the average of the sequence is $$\frac{1^{st} term+ 2^{nd}}{2}=\frac{45+243}{2}=\frac{288}{2}=144$$
SUM = average * number of terms = 144*100 = 14,400

B

the average of the sequence is $$\frac{1^{st} term+ 2^{nd}}{2}=\frac{45+243}{2}=\frac{288}{2}=144$$
Here it,s not 2nd term(the 243) it will be last term, edit this, Thanks
_________________

New to the GRE, and GRE CLUB Forum?
Posting Rules: QUANTITATIVE | VERBAL

Questions' Banks and Collection:
ETS: ETS Free PowerPrep 1 & 2 All 320 Questions Explanation. | ETS All Official Guides
3rd Party Resource's: All In One Resource's | All Quant Questions Collection | All Verbal Questions Collection | Manhattan 5lb All Questions Collection
Books: All GRE Best Books
Scores: Average GRE Score Required By Universities in the USA
Tests: All Free & Paid Practice Tests | GRE Prep Club Tests
Extra: Permutations, and Combination
Vocab: GRE Vocabulary

Director  Joined: 22 Jun 2019
Posts: 517
Followers: 4

Kudos [?]: 103 , given: 161

Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
sbingi wrote:
sandy wrote:
The sequence A is defined by $$A_n = A_{n – 1} + 2$$ for each integer n ≥ 2, and $$A_1 = 45$$. What is the sum of the first 100 terms in sequence A?

(A) 243
(B) 14,400
(C) 14,500
(D) 24,300
(E) 24,545

Solution:

Given A_1 = 45
n ≥ 2,
A_2 = A_1 + 2 = 45+2 = 47

Series {45, 47, 49.....)
First term a = 45 and common difference d = 2
Number of terms = 100

Sum = (n/2)*(2a + (n-1)d)
= (100/2) * (2 x 45 + 99 x 2)
= (100/2) * 288
= 14400

Avoid as much as possible the previous rules that we taught to memorize in School   _________________

New to the GRE, and GRE CLUB Forum?
Posting Rules: QUANTITATIVE | VERBAL

Questions' Banks and Collection:
ETS: ETS Free PowerPrep 1 & 2 All 320 Questions Explanation. | ETS All Official Guides
3rd Party Resource's: All In One Resource's | All Quant Questions Collection | All Verbal Questions Collection | Manhattan 5lb All Questions Collection
Books: All GRE Best Books
Scores: Average GRE Score Required By Universities in the USA
Tests: All Free & Paid Practice Tests | GRE Prep Club Tests
Extra: Permutations, and Combination
Vocab: GRE Vocabulary GRE Instructor Joined: 10 Apr 2015
Posts: 3823
Followers: 148

Kudos [?]: 4464  , given: 69

Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
2
KUDOS
Expert's post
sandy wrote:
The sequence A is defined by $$A_n = A_{n – 1} + 2$$ for each integer n ≥ 2, and $$A_1 = 45$$. What is the sum of the first 100 terms in sequence A?

(A) 243
(B) 14,400
(C) 14,500
(D) 24,300
(E) 24,545

Let's examine a few terms to see the pattern:
term1 = 45 (we add 0 two's)
term2 = 45 + 2 = 47 (we add 1 two)
term3 = 45 + 2 + 2 = 49 (we add 2 two's)
term4 = 45 + 2 + 2 + 2 = 51 (we add 3 two's)
.
.
.
term100 = 45 + 2 + 2 ....... + 2 = 243 (we add 99 two's)

So, the sum of the first 100 terms = 45 + 47 + 49 + . . . + 241 + 243
Let's add the values in PAIRS, by pairing up values from each side (left and right) of the sum.

That is: 45 + 47 + 49 + . . . + 239 + 241 + 243 = (45 + 243) + (47 + 241) + (49 + 239) + ....
= (288) + (288) + (288) + ....
Since we have 50 PAIRS that each add to 288, the TOTAL sum = (50)(288) = 14,400

Cheers,
Brent
_________________

Brent Hanneson – Creator of greenlighttestprep.com
If you enjoy my solutions, you'll like my GRE prep course. Manager Joined: 22 May 2019
Posts: 58
Followers: 0

Kudos [?]: 30 , given: 194

Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
sandy wrote:
Explanation

The first term of the sequence is 45, and each subsequent term is determined by adding 2. The problem asks for the sum of the first 100 terms, which cannot be calculated directly in the given time frame; instead, find the pattern.

The first few terms of the sequence are 45, 47, 49, 51,… What’s the pattern? To get to the 2nd term, start with 45 and add 2 once. To get to the 3rd term, start with 45 and add 2 twice. To get to the 100th term, then, start with 45 and add 2 ninety-nine times:

$$45 + (2)(99) = 243.$$

Next, find the sum of all odd integers from 45 to 243, inclusive. To sum up any evenly spaced set, multiply the average (arithmetic mean) by the number of elements in the set. To get the average, average the first and last terms. Since $$\frac{45+243}{2}= 144$$, the average is 144.

To find the total number of elements in the set, subtract 243 – 45 = 198, then divide by 2 (count only the odd numbers, not the even ones): $$\frac{198}{2}= 99$$ terms.

Now, add 1 (to count both endpoints in a consecutive set, first subtract and then “add 1 before you’re done”). The list has 100 terms. Multiply the average and the number of terms:

$$144 \times 100 = 14,400$$

It has been explicitly said that the sum of the first 100 terms. Isn't it enough to move on instead of calculating the total number of terms in the above way?
Manager  Joined: 19 Nov 2018
Posts: 102
Followers: 0

Kudos [?]: 105 , given: 53

Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
Here is the formula for finding a specific term in an arithmetic sequence. I didn't do the whole problem, because everyone else already has. I just thought this part might have been glossed over a little.
Attachments daum_equation_1566334680442.png [ 98.83 KiB | Viewed 2580 times ] Re: The sequence A is defined by An = An – 1 + 2 for each intege   [#permalink] 20 Aug 2019, 12:52
Display posts from previous: Sort by

# The sequence A is defined by An = An – 1 + 2 for each intege  Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group Kindly note that the GRE® test is a registered trademark of the Educational Testing Service®, and this site has neither been reviewed nor endorsed by ETS®.