Mathematics, 24.07.2019 16:00 reginaldlegette

Find the nth term of the sequence: 14, 37, 72, 119, 178

Answers: 1

Show answers

Answers

Answer from: broyochey1

Check the forward differences: if a_i

is the

-th term, then the first forward difference of a_i

is $b_i=a_{i+1}-a_i$ .

$\begin{cases}a_1=14\\a_2=37\\a_3=72\\a_4=119\\a_5=178\end{cases}$

$\implies\begin{cases}b_1=27-14=23\\b_2=72-37=35\\b_3=119-72=47\\b_4=178-119=59\end{cases}$

The usefulness is this: if the sequence were arithmetic, then the forward differences would be constant. For example, if the sequence were $\{1,2,3,\ldots\}$ , we would see the differences to be $\{1,1,\ldots\}$ , and so the sequence would be growing linearly, or by an added constant.

In our case, we can compute the forward differences again (the second-order differences) until we find such a pattern. This time, we denote it by $c_i=b_{i+1}-b_i$ .

$\implies\begin{cases}c_1=35-23=12\\c_2=47-35=12\\c_3=59-47=12\end{cases}$

So now we know that the sequence b_n

is arithmetic with a common difference of 12 between its terms.

b_4=b_3+12=b_2+24=b_1+36

Now consider a_3=72

; we can write this term of the sequence in terms of the previous ones. We have

$35-23=12\implies 35=23+12$
$\implies72-37=23+12$
$\implies\underbrace{72}_{a_3}=\underbrace{37}_{a_2}+\underbrace{23}_{a_2-a_1=37-14}+12$
$\implies a_3=2a_2-a_1+12$

We would see a similar recursive pattern if we looked at the other terms a_4=119

and

. We then establish that the sequence is given by the recursive formula

$\begin{cases}a_1=23\\a_2=35\\a_{n+2}=2a_{n+1}-a_n+12&\text{for }n2\end{cases}$

There are lots of ways to find the explicit formula for a_n

from this point, but the simplest is to realize that the sequence must be quadratic (this is because the second-order differences are constant) so we can assume that

a_n=an^2+bn+c

for some constants a,b,c

. We have to solve for three unknowns, so we need three known values:

$n=1\implies14=a+b+c$
$n=2\implies37=4a+2b+c$
$n=3\implies72=9a+3b+c$

$\implies a=6,b=5,c=3$

So the

-th term is determined by

a_n=6n^2+5n+3

for all $n\ge1$ .

Answer from: Quest

take the top number and divide it by a the bottom number

step-by-step explanation:

1/2 1 divied by 2 = .5

Answer from: Quest

the central idea is that:

individuals should refuse to cooperate with institutions they believe unjust.

thoreau felt that governments were more often agents of corruption than justice, even democracies, and that just because the majority feels a certain way does not make it just or correct in absolute terms. he uses the fact that slavery exists in our democracy to support this idea.

he goes on to say that, as a result, people should live true to their own internal laws regarding right and wrong, as a product of reflection and a moral life, rather than following every law that society imposes.

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 12:40

Definition: es. this is a disadvantage or weak point that makes someone or something less effective.

Answers: 2

Answer

Mathematics, 21.06.2019 17:40

Which of the following are accepted without proof in a logical system? check all that apply.

Answers: 1

Answer

Mathematics, 21.06.2019 18:30

In the following diagram it is given that dec,ab || dc, ad || eb, and ad is congruent to bc. a)why isn't abc a parallelogram even though it has a pair of parallel sides and a pair of congruent sides b)explain why be must be congruent to bc. further explain what type of triangle this makes triangle abc and what it tells you about angle 1 and angle 2c) finally why must angle 3 be congruent to angle 1? further, explain why we know that angle 3 is congruent to angle 2

Answers: 1

Answer

Mathematics, 21.06.2019 19:30

Піf x = y and y= z, which statement must be true? оа. -x = -2 b. x= z ос. —x = 2 od. 2= x

Answers: 3

Answer

You know the right answer?

Find the nth term of the sequence: 14, 37, 72, 119, 178...

Questions