BAM! Mr. Tarrou In this video we're gonna take a look at a really fun question
involving arithmetic sequences and you might go but come on mr. true they're so
easy they just generally give us all the numbers and the answers pop out we just
use a formula for the general term and the partial sum same with the geometric
sequences but this time we're have to dig in our toolbox and use multiple
skills at the same time this is written and inspired by a type of question that
you may see on an IB math exam or just an interesting exam a fun exam something
maybe like a bonus question or just you know involving more than one or two
skills at the same time so we have here in an arithmetic sequence the sum of the
third and seventh terms is two and given the sum of the first eight terms as 24
determine the first term and the common difference how often do you see
questions about arithmetic and geometric sequences where they don't hardly tell
you anything we don't know a common difference we don't have even any
specific terms to build off of we just know that the sum of the third and the
seventh is two and the sum of the first eight is 24 well now the sum of the
first eight that sounds like a purchase sum formula I guess you could kind of
argue excuse me for all the noise I guess you could argue that they're both
partial sums but we have a formula to help us with the first eight so let's
not forget that a geometric twos mean an arithmetic sequence is a just basically
a list of numbers that are created with constant addition your general term is a
sub N and it's just that that value that's counting along the index are you
looking to first the second the third number or so on a sub n is equal to a
sub 1 the first term plus and minus 1 times D your common difference may be
you're constantly adding by 2 or 5 or minus 7 every time or some kind of
fractional value partial sum the sum of the first so many terms is equal to n
over 2 times a sub 1 plus a sub n we're going to rely on these formulas and our
confidence in our math skills too you work for this problem and if we
don't have any concrete numbers right away like it's not just you can't just
glance at and go oh a sub 1 is 10 that's okay just keep going know what you've
written is correct fill in the formulas that is involved in
this question like the sum of the third and the seventh term is two why don't we
just set up a formula for adding those two and set it equal to two the third
term in the sequence not the first third term in the sequence is going to be
equal to the first term plus n which is going to be three for a sub three so
three minus one times the common difference so a sub 3 is equal to a sub
1 plus 2 d c-- we know what a sub 3 is it's whatever that is a sub 7 well a sub
7 is going to be a sub 1 plus 7 minus 1 times D a sub 1 plus 60
find it kind of hard write neatly this morning for some reason now when you
take these two terms a sub 3 and a sub 7 and add them it needs to be 2 so okay we
don't really know what these the 3rd and 7th term or number and the arithmetic
sequence is but when we add them a sub 1 plus 2 D plus a sub 1 + 6 D that needs
to come out to be 2 combining like terms we have 2 times a sub 1 + 8 D is equal
to 2 divide everything by 2 correctly unlike the first attempt to shoot this
video and we have a sub 1 + 2 8 divided by 2 is 4 D
and make sure you divide both sides there we go
made that mistake I wrote it as to a minute ago now this is going to be some
again it's it's an equation I mean I still don't know what any concrete
numbers are I don't know what the first term is nor do I know yet the common
difference but I have some relation between those two that I've created from
the fact that the 3rd and 8th 7th excuse me
term add up to 2 now let's turn our attention with a different color to the
fact that the sum of the first eight terms is 24 see what we get off of that
the sum of the first eight terms is going to be equal to well now we're
going to use that partial sum formula and eight over two and what is the
partial sum formula asking for the first term so we don't know and the nth term
or in this case the eighth term the a sub eight term well I don't really have
the ace of eighth term so we're have to figure that out just like we found out
the a sub three and a sub seventh term and get the phone
so trying that again we have a sub eight is equal to a sub one times n minus one
eight minus one plus I said x plus seven D okay so a sub eight is a sub 1 plus 7
d + a sub 8 is equal to 24 so we have 24 is equal to what four times a plus a is
2 a sub 1 + 7 D divide both sides by 4 and we get 6 is equal to 2 a sub 1 + 7 D
great well I still don't necessarily know much but we're getting close we
have a formula relating a sub 1 and D
together based on the sum of the first eight terms it's right here in purple
we've got a relation between a sub 1 and d 4 when we add the third and the
seventh term this is now getting into something you learned back in algebra 1
it's a linear system and you solve a two variable linear system by either doing
linear combination or substitution I'm going to go ahead and pick substitution
because here we have a nice neat a sub 1 is equal to 1 minus 4 D let's go ahead
and plug this in to a sub 1 because and we can finally we can find the a or a
sub 1 or D first doesn't matter but this is going to be a cleaner substitution
that's going to give us that 6 is equal to 2 times a sub 1 which is 1 minus 4 D
plus 7 D that's going to give us after we distribute by 2 and combine like
terms 6 is equal to 2 minus 8 D plus 7 D we have 4 after we subtract by 2 is
equal to negative 8 plus 7 is negative 1 D and negative 4 is our common
difference well now that we know what our common difference is we can go up
here and say that well using this relationship here that a sub 1 is equal
to 1 minus 4 times D which is negative 4 that's 1 plus 6 16 a sub 1 is equal to
17 now if you want to double-check this it's really simple you just start
listing off and creating those terms a sub 1 and 17 a sub 2 is 13 because
you're subtracting by 4 okay I'm going to run out of space if I keep writing a
sub 1 a sub 2 a sub 3 but we have 17 13 9 take away 4
one negative three negative seven one two three four five six seven negative
eleven now you can add up these terms and make sure that you get the 24 that
it is add the third and the four five six seven term we have 9 plus negative
seven is 2 and we have checked our answer I'm mr. Tarrou BAM go do that
homework
you