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2, 8, 14, 20

The very first term in the succession is 2, and each following term is determined by including 6. What is the value of the 50th term?

 

 


Explanation:

We begin by multiplying 6 times 46, due to the fact that the first 4 state are already listed. We then add the product, 276, come the last noted term, 20. This gives us our answer that 296.

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In one arithmetic sequence, every term is two better than the one that precedes it. If the sum of the very first five regards to the succession is equal to the difference in between the first and 5th terms, what is the tenth hatchet of the sequence?


Explanation:

Let a1 represent the first term the the sequence and also an represent the nth term.

We room told the each hatchet is two higher than the ax that comes before it. Thus, we deserve to say that:

a2 = a1 + 2

a3 = a1 + 2 + 2 = a1 + 2(2)

a4 = a1 + 3(2)

a5 = a1 + 4(2)

an = a1 + (n-1)(2)

The problem tells us that the sum of the very first five state is same to the difference in between the fifth and very first terms. Let"s compose an expression because that the amount of the very first five terms.

sum = a1 + (a1 + 2) + (a1 + 2(2)) + (a1 + 3(2)) + (a1 + 4(2))

= 5a1 + 2 + 4 + 6 + 8

= 5a1 + 20

Next, we desire to compose an expression because that the difference in between the fifth and very first terms.

a5 - a1 = a1 + 4(2) – a1 = 8

Now, we set the 2 expressions equal and solve because that a1.

5a1 + 20 = 8

Subtract 20 native both sides.

5a1 = –12

a1 = –2.4.

The question eventually asks us for the tenth ax of the sequence. Now, that we have the very first term, us can find the tenth term.

a10 = a1 + (10 – 1)(2)

a10 = –2.4 + 9(2)

= 15.6

The answer is 15.6 .


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Example concern #4 : Nth term Of an Arithmetic succession


In a particular sequence, an+1 = (an)2 – 1, wherein an represents the nth term in the sequence. If the 3rd term is equal to the square that the an initial term, and all of the terms room positive, then what is the value of (a2)(a3)(a4)?


Possible Answers:

24


63


48


6


72


Correct answer:

48


Explanation:

Let a1 it is in the first term in the sequence. We have the right to use the truth that an+1 = (an)2 – 1 in stimulate to uncover expressions for the second and third terms the the sequence in regards to a1.

a2 = (a1)2 – 1

a3 = (a2)2 – 1 = ((a1)2 – 1)2 – 1

We can use the reality that, in general, (a – b)2 = a2 – 2abb2 in bespeak to leveling the expression because that a3.

a3 = ((a1)2 – 1)2 – 1

= (a1)4 – 2(a1)2 + 1 – 1 = (a1)4 – 2(a1)2

We room told that the third term is same to the square of the very first term. 

a3 = (a1)2

We can substitute (a1)4 – 2(a1)2 for a3.

(a1)4 – 2(a1)2 = (a1)2

Subtract (a1)2 native both sides.

(a1)4 – 3(a1)2 = 0

Factor out (a1)2 from both terms.

(a1)2 ((a1)2 – 3) = 0

This method that either (a1)2 = 0, or (a1)2 – 3 = 0. 

If (a1)2 = 0, climate a1 should be 0. However, we room told that all the terms of the sequence are positive. Therefore, the an initial term can"t it is in 0.

Next, let"s solve (a1)2 – 3 = 0.

Add 3 to both sides.

(a1)2 = 3

Take the square root of both sides. 

a1 = ±√3

However, due to the fact that all the terms space positive, the only possible value because that a1 is √3.

Now, that we recognize that a1 = √3, us can uncover a2, a3, and also a4.

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a2 = (a1)2 – 1 = (√3)2 – 1 = 3 – 1 = 2

a3 = (a2)2 – 1 = 22 – 1 = 4 – 1 = 3

a4 = (a3)2 – 1 = 32 – 1 = 9 – 1 = 8

The question ultimately asks because that the product the the a2, a3, and also a4, which would certainly be same to 2(3)(8), or 48.