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GRE Prep Club Legend  Joined: 07 Jun 2014
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GRE 1: Q167 V156 WE: Business Development (Energy and Utilities)
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The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
Expert's post 00:00

Question Stats: 42% (01:07) correct 57% (01:14) wrong based on 7 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

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Sandy
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GRE Prep Club Legend  Joined: 07 Jun 2014
Posts: 4857
GRE 1: Q167 V156 WE: Business Development (Energy and Utilities)
Followers: 105

Kudos [?]: 1783 , given: 397

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

$$144 \times 100 = 14,400$$
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Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
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.
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Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
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
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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
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Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink]
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 Re: The sequence A is defined by An = An – 1 + 2 for each intege   [#permalink] 06 Jan 2019, 10:28
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