The sequence of numbers a1, a2, a3, , an, is defined by an = (n + 1

Question

The sequence of numbers  a_1 ,  a_2 ,  a_3 , …,  a_n , … is defined by  a_n = (n + 1)^2 − n^2  for each integer  n ≥ 1 . If the sum of the first x terms of the sequence is 80, what is the value of x ?

A. 8
B. 9
C. 10
D. 11
E. 12

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8.toastcrunch · Answer

The sum of the first x terms of the sequence is given by S = a1 + a2 + a3 + ... + ax. We can simplify the formula for the nth term as a_n = 2n + 1. Substituting this into the formula for S, we get:

S = a1 + a2 + a3 + ... + ax
 = (2(1) + 1) + (2(2) + 1) + (2(3) + 1) + ... + (2x + 1)
 = 2(1 + 2 + 3 + ... + x) + x

Using the formula for the sum of the first x positive integers, we get:

S = x(x + 1) + x
 = x^2 + 2x

Setting this equal to 80 and solving for x, we get:

x^2 + 2x = 80
x^2 + 2x - 80 = 0
(x + 10)(x - 8) = 0

The only positive solution is x = 8. Therefore, the value of x is 8.
Answer is option A.

hinasingh · Answer

The sequence of numbers a1, a2, a3, …, an, … is defined by an=(n+1)^2−n^2 for each integer n≥1. If the sum of the first x terms of the sequence is 80, what is the value of x ?

an = (n+1)^2 -n^2 = 2n+1
a1 = 3
a2 = 5
a3 =7 and so on

Using the formula for sum of first k odd integers (= k^2)
we know that, for k = 9, sum = 81
if we remove the first term 1, the sum is 80 and the number of terms is 8.
Answer A

youcouldsailaway · Answer

Started by simply substituting 8 in place of n and finding an for a few values (n = 8,7,6...). Once you see that the pattern of the differences is basically consecutive odd numbers, you can use that to find the n that can lead to the sum of the differences being 80, which in this case is 8.

szcz · Answer

In this kind of question we would be better of not opening the brackets as almost all the terms will easily get cancelled out.

a1= (2)^2-(1)^2
a2= (3)^2-(2)^2
a3= (4)^2-(3)^2
.
.
.
a8= (9)^2-(8)^2

On adding all the above terms, only 2 values will be left.
Total of 8 terms= 9^2-1^2=80
Hence, x=8.

Adit_ · Answer

Ws trying to think of the best approach to such questions:
On thinking, can we eliminate b,d,e as they aren't factors of 80?

In this case the median is going to be an even number and product of median*no.of items = sum of an AP thus I think using POE, B,D,E is gone.
Coming to A and C:
Taking 10 first:
The sequence is going like 3,5,7,9...
And 10*8 = 80, thus we need median as 8.
But we see that median is 8 when its just 6 numbers and 8*6=48 not 80.
Thus  8(A)  is our answer.

_______________________

With any POE I think this especially becomes faster considering in this case atleast we can check for A right away and it does work.

The sequence of numbers a1, a2, a3, , an, is defined by an = (n + 1

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The sequence of numbers a1, a2, a3, , an, is defined by an = (n + 1

Prep Toolkit

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