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

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GMAT Club Legend
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Joined: 07 Jun 2014
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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:

25% (01:19) correct 75% (01:07) wrong based on 4 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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GMAT Club Legend
GMAT Club Legend
User avatar
Joined: 07 Jun 2014
Posts: 4749
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 93

Kudos [?]: 1653 [0], given: 396

CAT Tests
Re: The sequence A is defined by An = An – 1 + 2 for each intege [#permalink] New post 12 Aug 2018, 04:45
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] 12 Aug 2018, 04:45
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