Good day students in this clip were going to be going over an example on finding arithmetic
sequence is of finding the
common difference to end term and the explicit formula of terms of an arithmetic sequence
so let's take a look at the of
instructions for the examples the sport the given sequence determine if it is an arithmetic
or geometric sequence find the
common difference for racial find the next three terms of the sequence find the value
of a thirty and lastly, find the
explicit formula is it to verify your answer indeed.
Right let's take a look at thus sequence under consideration for
number one we of fourteen four negative six and negative sixty in in the sequence continues
okay of the difference between
sequence a series is the elements of of the terms in a sequence are separated by commas
present a series you have plus or
minus is between them.
Okay so think of sequence against a list of numbers were asked to misuse represents the
Psalm of of
numbers okay right so am part payment to determine if it's arithmetic or geometric.
So when is negative arithmetic
registered a side note here arithmetic we have a common difference, and difference for
arithmetic unit is repeated
addition or subtraction that's how subsequent terms of the sequence are generated on the
other hand for geometric you have
a common ratio okay, one graciously constantly multiply to generate of the next terms of
the geometric sequence to think
of this as pattern subtract and this would think of it as multiply and divide right so
determine if his arithmetic or
geometric the going to see if our differences the ratio it will, and difference or the common
ratio okay so let us start
with arithmetic tests first for arithmetic series to go a different D has to be equal
to of eight two minus eighty-one
asked to be equal to the three minus eight two and has equal to a four minus eight three
and out pattern have to continue
basically the formula for, differences a seven and a term minus the term before it is how
you guessed it, and difference
team is get the same number every single time.
Right let's look at the series in this case of this is a one the first time
the index tells you the position okay from a one and is the last terms to that's a one
and a half eight two and this is a
three in this term have right here is a four okay so let's use the services see if we're
having a common difference here
okay if it's not a common difference that this is not an arithmetic sequence right is
apply the formula eight two minus
eighty-one eighty minus eighty-one cases for minus eighty-one is fourteen okay and then
what is of this coming what is
this cannot to four minus fourteen is negative ten right let's try a three minus eight two
eight three is negative six
minus eight to negative four.
The computer difference have negative ten looking get a four minus eight three A4 is
negative sixteen minus of negative six we it becomes negative sixteen plus sixty go/--
these two basic into six three to
number negative be subtract and keep the number of the the sign of the bigger number so this
is negative ten also what is
the about this three numbers the constantly the same so this means that this is an arithmetic
sequence this is an
arithmetic sequence sequence line of because we have a common difference okay because there
is a common difference there
is a common difference right of another way to look at it is look at the pattern of numbers
right here from 10 to 4
subtract ten from four to negative six is subtract ten from negative sixteen negative
sixteen subtract ten to that's a cup
of the of the pattern have a common different happening over and over again.
Right now let's take a look at the be part a
says find a common difference our ratio is with determined that this is an arithmetic
sequence the validity of ratio is
not applicable here.
If this were a geometric sequence and we looking for the ratio okay but this is a an arithmetic
sequence we looking for the con different D and that's W the formula a sub and minus
case of an minus one this formula
simply means that you subtract the term from attorney for to get the common difference
right so we can pick any two terms
with it but there has to be a term of of any two terms will suffice okay so how about we
use him and equals three so how
will that how this apply to this formula to find a common difference of eight three of
the reason eight three minus eight
two okay and minus one three minus one is two eight three is negative six minus eight
two which is for the common
difference is negative ten right so that's that I want anyway notes is in the common
difference is negative, it causes the
sequence to reduce in size okay so that's what's happening here so the sign is actually
very important of and we also knew
that the so the answer right because we are there to determine area that were constantly
taken out ten from every term of
our sequence.
Right let's move on to part C this is for the next three terms of the sequence of so
let's rewrite the
sequence of part C the first term.
Anyone is fourteen the second term eight two were given was for a three is negative sex
A4 is negative sixteen asked to find the next three terms.
So let's look for a five how to find a five to find if I will
simply subtract the common difference from the term before it right so negative sixteen
minus ten or leading technical is
the adding negative ten sink in a subtract and write to negative sixty minus ten is equal
to negative twenty-six and in a
six how the find a six you start from the term before it negative twenty-six which is
a five and you subtract the common
difference again to that end grain subtract the common difference again that gives you
the next term minus ten equals
negative thirty-six and a seven the seventh term the take the term before it negative
thirty-six subtract the common
difference a at the common difference as the common difference negative thirty-six minus
tan is negative forty-six right
so next three terms next three terms are negative twenty-six negative thirty-six and negative
forty-six these correspond
two terms five six and seven so the index basically tells you the position right is
go to the next parts part D so for
part D we have to find of a thirty okay to find is of thirty now what on earth does this
mean this means that were to find
the value of the thirtieth third term okay so of what is the formula for finding the
end term of the arithmetic sequence
to the formula for that the phone was a and equals A1 the starting term plus and minus
one number terms minus one times
the common difference to that is the formula.
The going to be using is another way of doing it where you can just simply
of cheap for subtracting your common difference audio common difference repeatedly in a get
the end term but this is facet
as it is you can jump from the first term into your F term by taking advantage of the
multiplicative nature of the common
difference right so let's go ahead and use formula of a thirty this case integral to
that and an is the index okay, it's
always the index so and equals thirty A1 is the first term of the sequence which is fourteen
D is a common difference were
to determine that that is of negative ten to the plug all these values into our formula
right here so is going to have a
sub thirty is equal to anyone which is fourteen plus and minus one the thirty minus one times
the common difference which
is negative ten the equals now is this expression we just simplify using order of operations
right to this is my dear aunt
Sally so do the parenthesis first thirty minus one is twenty-nine times negative ten next
operation is multiplication so
the have fourteen minus two hundred and ninety and then of fourteen minus two ninety is equal
to negative two hundred and
seventy-six and that's your thirtieth third term of your sequence okay knots move on to
part E to part B of we are to find
the explicit formula in a verify the accuracy of times and D using that of formula okay
so far X is it formula for the
sequence is basically us simplify in the end term formula for any arbitrary term in okay
so the form of our and sterling
is a and is equal to A1 plus and minus one D so all you simply do is of plug-in A1 in
the and that helps to find the
formula for the end term okay so in this case and is equal to so what and write down the
only difference between this and
part D and is equal to and A1 will assume the Valley of A1 which is fourteen D is going
to be negative ten right so you
see the difference between these two this has a specific and value which is thirty but
we looking for the explicit formula
this is how you you do it okay so remember we looking for the end the explicit formula
for the end term right so of let's
go ahead and simplify this in the have a and equals the first term which is fourteen plus
and minus one in is to and one
is one of the con difference of negative ten simplify this will have fourteen of plus now
just simply distribute this ten
to these two terms right here distributed to have of negative ten and minus ten minus
one is plus ten okay plus of the
five further fourteen plus times minus is minus minus ten ten plus ten right is go my
like terms this in this are like
terms the constants so can combine them into twenty-four minus ten and so this formula
is the explicit formula for the
instant explicit formula right okay so of we of how the X this is formula is it right
or is it rugged correct now what
were going to do the going to use his explicit formula to find the thirty X term and see
if we end up with the same answer
of negative to seventy-six okay so what we do.
Right now is a test okay if it's wrong if we deny get this answer them is
either X is it formula is wrong or our computation in part, these run of both of them on okay
so hopefully get a right
answer that then we confident that we fact correct okay so let's of good to for the X
is it formula all you need to is the
end term okay so in this case and on the position and we need and is thirty so let's plug it
into of the X is formula is a
is of thirty is going to be equal to twenty-four minus ten times and which is thirty and leave
the let's simplify this is
the order of operations ten times thirty-three hundred and then three hundred minus twenty-four
of is negative two hundred
and seventy-six and that is perfect that is exactly what we had here so the confidence
that our answer the correct our X
is it formula and are and term result four of the house accurate right so that's that
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