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The sequence A is defined by An = An – 1 + 2 for each intege

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The sequence A is defined by An = An – 1 + 2 for each intege [#permalink] New post 27 Jul 2018, 01:31
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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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Joined: 07 Jun 2014
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Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink] New post 12 Aug 2018, 04:45
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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] New post 06 Jan 2019, 08:41
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] New post 06 Jan 2019, 08:56
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

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Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink] New post 06 Jan 2019, 10:28
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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