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# What is the smallest positive integer n, such that the sum of 2n terms

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Math Expert
Joined: 02 Sep 2009
Posts: 64949
What is the smallest positive integer n, such that the sum of 2n terms  [#permalink]

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15 May 2020, 07:19
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58% (01:38) correct 42% (02:30) wrong based on 10 sessions

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Sequence A = {1, 5, 9, 13, ...}
Sequence B = {56, 58, 60, 62, ...}

What is the smallest positive integer n, such that the sum of 2n terms of sequence A is greater than the sum of n terms of sequence B?

A. 8
B. 9
C. 10
D. 12
E. 14

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Re: What is the smallest positive integer n, such that the sum of 2n terms  [#permalink]

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15 May 2020, 07:45
Bunuel wrote:
Sequence A = {1, 5, 9, 13, ...}
Sequence B = {56, 58, 60, 62, ...}

What is the smallest positive integer n, such that the sum of 2n terms of sequence A is greater than the sum of n terms of sequence B?

A. 8
B. 9
C. 10
D. 12
E. 14

Seq_A: $$S_{2n} = \frac{2n(2+(2n-1)4)}{2}$$

Seq_B: $$S_{n} = \frac{n(2*56+(n-1)2)}{2}$$

$$n(8n-2) > \frac{n(112 + 2n -2)}{2}$$

$$8n^2-2n > n(n +55)$$
$$8n^2-2n > n^2 + 55n$$
$$7n^2 > 57n$$
$$7n > 57$$
n = 9
Ans: B
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Re: What is the smallest positive integer n, such that the sum of 2n terms  [#permalink]

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15 May 2020, 18:49
Sequence A = {1, 5, 9, 13, ...} = 4*m+1, m = 0, 1, 2, ......n,......2n
Sequence B = {56, 58, 60, 62, ...} = 2*m+56, m = 0, 1, 2, ......n,......2n

Sum of 2n terms in
$$A_{2n} = [4*1+1] + [4*2+1] + .......... [4*2n+1]$$
$$A_{2n} = 4[1+2+.....2n] + [1+1+.......1 (2n times)]$$
$$A_{2n} = 4[\frac{2n(2n+1)}{2}] + 2n$$
$$A_{2n} = 4n(2n+1)+2n$$

Sum of n terms in B
$$B_n = 2*m+56$$
$$B_n = [2*1+56] + [2*2+56]+......... [2*n+56]$$
$$B_n = 2*[1+2+.....n] + 56*n$$
$$B_n = 2*\frac{n(n+1)}{2}+56n$$

Now, A_2n > B_n
4n(2n+1)+2n >n(n+1)+56n
7n>51 ---> for n = 8, 7n is > 51
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Re: What is the smallest positive integer n, such that the sum of 2n terms  [#permalink]

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15 May 2020, 20:01
Bunuel wrote:
Sequence A = {1, 5, 9, 13, ...}
Sequence B = {56, 58, 60, 62, ...}

What is the smallest positive integer n, such that the sum of 2n terms of sequence A is greater than the sum of n terms of sequence B?

A. 8
B. 9
C. 10
D. 12
E. 14

Are You Up For the Challenge: 700 Level Questions

For sequence 1, a = 1, d =4.
For seq 2, a =56, d =2
We need to find, a + (2n -1) 4 > a +(n-1)2
or, 1 + 8n - 4 > 56 + 2n -2
or, 6n -3 > 54
or, 6n > 57
or, n > 9. something
so, the smallest possible integer n can take is 10.
Re: What is the smallest positive integer n, such that the sum of 2n terms   [#permalink] 15 May 2020, 20:01