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For each integer n>1, if S(n) denote the sum of even integer

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For each integer n>1, if S(n) denote the sum of even integer [#permalink] New post 25 Aug 2018, 12:40
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For each integer \(n>1\), if S(n) denote the sum of even integer upto \(n\) (not inclusive of \(n\)). For example, \(S(10)= 2+4+6+8=20\). What is value of \(S(300)\)?

(A) \(22050\)
(B) \(22350\)
(C) \(22650\)
(D) \(45150\)
(E) \(90300\)
[Reveal] Spoiler: OA

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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] New post 31 Aug 2018, 03:28
hi

can anyone tell whats the shortcut to such questions?
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] New post 31 Aug 2018, 04:28
Expert's post
IshanGre wrote:
hi

can anyone tell whats the shortcut to such questions?



You need to be well versed with arithmetic progression formulas. It is not super complex just needs practice
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] New post 31 Aug 2018, 05:40
IshanGre wrote:
hi

can anyone tell whats the shortcut to such questions?


First, you need to memorize the formula for the sum of arithmetic progression. Second, you need to know how to count number of terms. Third, practice and practice.
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] New post 31 Aug 2018, 21:45
please tell the shortcut method to solve this question.
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] New post 10 Nov 2018, 07:39
S(300) = 2+..........+298

for number of terms, l = a+(n-1)*2
298 = a+(n-1)*2 = 2+(n-1)*2 => 296/2 = n-1 => n=149

for sum,

S(300) = n/2 * (a+l) = (149/2)*(298+2) = 149*150 = 22350


Answer is B!

Last edited by indiragre18 on 11 Nov 2018, 00:16, edited 1 time in total.
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] New post 10 Nov 2018, 23:28
Expert's post
sandy wrote:
For each integer \(n>1\), if S(n) denote the sum of even integer upto \(n\) (not inclusive of \(n\)). For example, \(S(10)= 2+4+6+8=20\). What is value of \(S(300)\)?

(A) \(22050\)
(B) \(22350\)
(C) \(22650\)
(D) \(45150\)
(E) \(90300\)



there are three ways to do it ....

(I) If you know that Sum of first n integers is \(\frac{n(n+1)}{2}\)
Sum = \(2+4+6+...+300) = 2(1+2+3....+150)= 2 *\frac{150*151}{2}=150*151=22650\)

(II) If you know that Sum of first n integers is \(\frac{n(n+1)[}{fraction]\)
Now we have \([fraction]300/2}=150\) terms till 300, inclusive.
Sum = \(2+4+6+...+300 = 150*151=150*151=22650\)

(III) since it is an AP. the sum will be equal to Number of integers* average
so \(150 * \frac{(300+2)}{2} = 150*151 = 22650\)

Now subtract 300 from each result as the answered has to be exclusive of 300, that is 300 is not be included in total..
Therefore answer is 22650-300=22350

B

To know more about Arithmetic progressions
https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
_________________

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: For each integer n>1, if S(n) denote the sum of even integer [#permalink] New post 26 Nov 2018, 14:19
chetan2u wrote:
sandy wrote:
For each integer \(n>1\), if S(n) denote the sum of even integer upto \(n\) (not inclusive of \(n\)). For example, \(S(10)= 2+4+6+8=20\). What is value of \(S(300)\)?

(A) \(22050\)
(B) \(22350\)
(C) \(22650\)
(D) \(45150\)
(E) \(90300\)



there are three ways to do it ....

(I) If you know that Sum of first n integers is \(\frac{n(n+1)}{2}\)
Sum = \(2+4+6+...+300) = 2(1+2+3....+150)= 2 *\frac{150*151}{2}=150*151=22650\)

(II) If you know that Sum of first n integers is \(\frac{n(n+1)[}{fraction]\)
Now we have \([fraction]300/2}=150\) terms till 300, inclusive.
Sum = \(2+4+6+...+300 = 150*151=150*151=22650\)

(III) since it is an AP. the sum will be equal to Number of integers* average
so \(150 * \frac{(300+2)}{2} = 150*151 = 22650\)

C

To know more about Arithmetic progressions
https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048


According to the initial problem, the answer is B. Also it seems you are doing inclusive of n=300, while the prompt states NON-inclusive.
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] New post 26 Nov 2018, 18:05
Expert's post
projectoffset wrote:
chetan2u wrote:
sandy wrote:
For each integer \(n>1\), if S(n) denote the sum of even integer upto \(n\) (not inclusive of \(n\)). For example, \(S(10)= 2+4+6+8=20\). What is value of \(S(300)\)?

(A) \(22050\)
(B) \(22350\)
(C) \(22650\)
(D) \(45150\)
(E) \(90300\)



there are three ways to do it ....

(I) If you know that Sum of first n integers is \(\frac{n(n+1)}{2}\)
Sum = \(2+4+6+...+300) = 2(1+2+3....+150)= 2 *\frac{150*151}{2}=150*151=22650\)

(II) If you know that Sum of first n integers is \(\frac{n(n+1)[}{fraction]\)
Now we have \([fraction]300/2}=150\) terms till 300, inclusive.
Sum = \(2+4+6+...+300 = 150*151=150*151=22650\)

(III) since it is an AP. the sum will be equal to Number of integers* average
so \(150 * \frac{(300+2)}{2} = 150*151 = 22650\)

C

To know more about Arithmetic progressions
https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048


According to the initial problem, the answer is B. Also it seems you are doing inclusive of n=300, while the prompt states NON-inclusive.



Yes, thank you.
I had included 300 in each case.
B will be the correct answer.
_________________

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

Re: For each integer n>1, if S(n) denote the sum of even integer   [#permalink] 26 Nov 2018, 18:05
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