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# If a sequence is given by the expression S_n = S_{n-1} + 3 and S_1 = 1

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Math Expert
Joined: 02 Sep 2009
Posts: 46305
If a sequence is given by the expression S_n = S_{n-1} + 3 and S_1 = 1 [#permalink]

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17 Sep 2017, 03:20
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45% (medium)

Question Stats:

74% (01:16) correct 26% (01:25) wrong based on 62 sessions

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If a sequence is given by the expression $$S_n = S_{n-1} + 3$$ and $$S_1=1$$, what is the sum of the first 30 terms of the series?

A. 1335
B. 885
C. 465
D. 88
E. 58

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Re: If a sequence is given by the expression S_n = S_{n-1} + 3 and S_1 = 1 [#permalink]

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17 Sep 2017, 07:43
Let the first term of the sequence be x.
Second term will be x+3, third term is x+3+3

So, the sum will be of form
30x + 3 + 6 + 9 + ....... 42 + 45 + 48 + ...... + 87
= 30x + 90*14 + 45
= 30x + 1305

If x = 1, the sum is 1335(Option A)
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Re: If a sequence is given by the expression S_n = S_{n-1} + 3 and S_1 = 1 [#permalink]

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17 Sep 2017, 10:06
1
Is this question complete???
I mean I am confused ..because we need atleast one term to get the exact answer.. be it the first term or any see term.

Kindly bear with me for my misunderstanding.

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Math Expert
Joined: 02 Sep 2009
Posts: 46305
Re: If a sequence is given by the expression S_n = S_{n-1} + 3 and S_1 = 1 [#permalink]

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17 Sep 2017, 10:10
kumarparitosh123 wrote:
Is this question complete???
I mean I am confused ..because we need atleast one term to get the exact answer.. be it the first term or any see term.

Kindly bear with me for my misunderstanding.

Sent from my Lenovo TAB S8-50LC using GMAT Club Forum mobile app

Edited the question: $$S_1=1$$ part was missing. Thank you for noticing it.
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Re: If a sequence is given by the expression S_n = S_{n-1} + 3 and S_1 = 1 [#permalink]

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17 Sep 2017, 10:21
2
1
S1 = 1,
S2 = S1+3
S3 = S2+3
.
.
.
.
.
.
.
.
S30= S29+3

so.. these terms are in A.P with the difference of 3

Lets find Last term S30 = S1 + (n-1)d
S1=1, n=30, d=3... solving we get S30 = 88

Sum of terms in A.P = (first term + Last term) * Number of terms/2
Sum = (1+88)* 30 /2
= 89*15
=1335

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If a sequence is given by the expression S_n = S_{n-1} + 3 and S_1 = 1 [#permalink]

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19 Sep 2017, 07:38
1
Bunuel wrote:
If a sequence is given by the expression $$S_n = S_{n-1} + 3$$ and $$S_1=1$$, what is the sum of the first 30 terms of the series?

A. 1335
B. 885
C. 465
D. 88
E. 58

$$S_n = S_{n-1} + 3$$ and
$$S_1=1$$

$$S_2 = S_{n-1} + 3$$

$$S_2 = S_{1} + 3$$ = 4
$$S_3 = S_{2} + 3$$ = 7

OR
$$S_1 = 1$$
$$S_2 = 1 + 3$$
$$S_3 = 1 + 3 + 3$$
$$S_4 = 1 + 3 + 3 + 3$$

For $$S_{n}$$, then, the first term is $$S_1 = 1$$, and the number of 3s is one fewer than $$n$$. It's an arithmetic series, first term is 1, common difference of 3, such that

$$S_n = S_1 + 3(n-1)$$

$$S_{30} = 1 + (29)3 = 88$$

Sum of arithmetic series is (average) * (number of terms), where average is (First Term + Last Term)/2

$$\frac{(1+88)}{2}$$ * (30) = (89)(15) = 1,335

No need to do the last calculation. Round 89 up to 90: (90)(15). Halve and double:(45)(30) > 1200. Only A is close.

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Re: If a sequence is given by the expression S_n = S_{n-1} + 3 and S_1 = 1 [#permalink]

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19 Sep 2017, 12:14
Bunuel wrote:
If a sequence is given by the expression $$S_n = S_{n-1} + 3$$ and $$S_1=1$$, what is the sum of the first 30 terms of the series?

A. 1335
B. 885
C. 465
D. 88
E. 58

$$S_n = S_{n-1} + 3$$

The above can be written as $$S_n - S_{n-1} = 3$$

The conclusion is that this is an AP series, whose first term is 1 and common difference is 3.

Sum of an AP series = $$\frac{n}{2}*[2a + (n-1)*d]$$ = $$\frac{30}{2}*[(2*1 + (30-1)*3]$$ = 1335

Thus, the correct answer is Option A.
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Re: If a sequence is given by the expression S_n = S_{n-1} + 3 and S_1 = 1   [#permalink] 19 Sep 2017, 12:14
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