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The sequence A is defined by An = An – 1 + 2 for each integer n >= 2

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The sequence A is defined by An = An – 1 + 2 for each integer n >= 2  [#permalink]

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New post 30 Jul 2018, 00:44
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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

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

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New post 30 Jul 2018, 02:03
\(A_ n = A_{n-1} + 2\)

\(A_ 1 = 45\)

\(A_ 2 = A_ 1 + 2(1)\)

\(A_ 3 = A_ 2 + 2 = A_ 1 + 2 + 2 = A_ 1 + 2(2)\)

Similarly \(A_ n = A_ 1 + (n-1)(2)\)

=> \(A_1 = 45\), \(A_2 = 47\), ... \(A_{100} = 45 + 99(2) = 243\)

Sum of first 100 terms = \(A_1 + A_2 + A_3 + .... A_{100}\)

=> 45 + 47 + 49 + ..... + 243

=> (45+0) + (45+2) + (45+4) + ... + (45+198)

=> (45*100) + (0+2+4+.. +198)

=> 4500 + 2(1+2+3+..+99)

=> 4500 + 2(50*99)

=> 4500 + (100*99)

=> 4500 + 9900

=> 14400

Hence option B
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Re: The sequence A is defined by An = An – 1 + 2 for each integer n >= 2  [#permalink]

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New post 30 Jul 2018, 02:24
First term = 45
2nd = 47 , 3rd term = 49. It's an AP of 2.
So last term = 45 + (100-1) 2 = 243
Sum of first 100 terms = (Average of first and last term)* no of terms = ((45+242)*100)/2 = 14400
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The sequence A is defined by An = An – 1 + 2 for each integer n >= 2  [#permalink]

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New post 30 Jul 2018, 03:23
Bunuel 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



Each term is 2 more than the previous term, so it is a arithmetic progression and in AP, average is the middle number and sum is average*number of terms

Here there are 100 terms, so our average will be middle of 50th and 51st ..
50th term = 45+(50-1)*2 =45+49*2 and
51st term = 45+(51-1)*2=45+50*2=145
Calculation of 51st is simple so let us save time with that

The middle of 50th and 51st will be 1 less than 51st as difference between the two is 2..
So 145-1=144
Sum=144*100=14400
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