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In a certain sequence, the term an is defined by the formula

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In a certain sequence, the term an is defined by the formula [#permalink] New post 27 Jul 2018, 07:15
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In a certain sequence, the term \(a_n\) is defined by the formula \(a_n = a_{n – 1} + 5\) for each integer \(n ≥ 2\). If \(a_1 = 1\), what is the sum of the first 75 terms of this sequence?

(A) 10,150
(B) 11,375
(C) 12,500
(D) 13,950
(E) 15,375
[Reveal] Spoiler: OA

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Re: In a certain sequence, the term an is defined by the formula [#permalink] New post 12 Aug 2018, 05:33
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Explanation

This is an arithmetic sequence: each new number is created by adding 5 to the previous number in the sequence.

Calculate the first few terms of the sequence: 1, 6, 11, 16, 21, and so on.

Arithmetic sequences can be written in this form: \(a_n = a_1 + k(n - 1)\), where k is the added constant and n is the number of the desired term. In this case, the function is: \(a_n = 1 + 5(n - 1)\).

The 75th term of this sequence is \(a_{75} = 1 + 5(74) = 371\).

To find the sum of an arithmetic sequence, multiply the average value of the terms by the number of terms.

The average of any evenly spaced set is equal to the midpoint between the first and last terms.

The average of the 1st and 75th terms is \(\frac{1+371}{2}= 186\). There are 75 terms. Therefore, the sum of the first 75 terms \(= 186 \times 75 = 13,950\).
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Re: In a certain sequence, the term an is defined by the formula [#permalink] New post 06 Jul 2019, 00:41
sandy wrote:
In a certain sequence, the term an is defined by the formula an = an – 1 + 5 for each integer n ≥ 2. If \(a_1 = 1\), what is the sum of the first 75 terms of this sequence?

(A) 10,150
(B) 11,375
(C) 12,500
(D) 13,950
(E) 15,375


any other way to solve? Please but not like Manhattan 5lb.
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Re: In a certain sequence, the term an is defined by the formula [#permalink] New post 08 Jan 2020, 17:14
Expert's post
In this kind of problems the trick is to find a pattern.

\(a_1=1\)
\(a_2=1+5=6\)
\(a_3=6+5=11\)
\(a_4=11+5=16\)
\(a_5=16+5=21\)
\(a_6=21+5=26\)
.......

If you sum the first two you do have 7, the third and the fourth you do have 27, and the fifth and the sixth 47

Notice how you do have an increment of 20 for each couple of numbers you add

if you extend the pattern til \(a_{10}\) you have 87

\(87 \times 75 = 6525 \times 2 \) (for the increment of 20, you add a couple of numbers which is two) \(\approx 13.050\)

Which is D, the closest answer choice
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Re: In a certain sequence, the term an is defined by the formula   [#permalink] 08 Jan 2020, 17:14
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