### Video Transcript

If the seventh term of an
arithmetic progression is one-ninth and its ninth term is one-seventh, find its 63rd
term.

Remember an arithmetic progression
is a sequence of numbers such that the difference between each of the consecutive
terms is constant. Let’s recall the formula to help us
find any term in the sequence. The 𝑛th term is 𝑎 plus 𝑛 minus
one multiplied by 𝑑, where 𝑎 is the first term and 𝑑 is the common
difference. We’ll first need to calculate the
value of the first term and the common difference.

To do so, let’s first substitute
what we know about the seventh term into the formula. The seventh term is one-ninth. So that gives us one-ninth is equal
to 𝑎 plus seven minus one 𝑑. Seven minus one is six. So we can simplify this expression
to give us one-ninth is equal to 𝑎 plus six 𝑑. Similarly, the formula for the
ninth term is one-seventh is equal to 𝑎 plus nine minus one multiplied by 𝑑. Nine minus one multiplied by 𝑑
simplifies to eight 𝑑.

Notice that we now have two
simultaneous equations. We can solve these to find a value
of 𝑎 and 𝑑. First, let’s subtract the equation
for the seventh term from the equation we created for the ninth term. This will eliminate the 𝑎 and give
us a single equation in terms of 𝑑. 𝑎 minus 𝑎 is zero and eight 𝑑
minus six 𝑑 is two 𝑑. So we get one-seventh minus
one-ninth is equal to two 𝑑.

To subtract these fractions, we’ll
find the lowest common denominator. The lowest common multiple of seven
and nine is 63. So the lowest common denominator of
these fractions is 63. To get 63 on the denominator of the
first fraction, we multiply by nine. We have to do the same to the
numerator. And nine multiplied by one is
nine.

Similarly, we multiply the
denominator of the second fraction by seven. So we have to do the same to the
numerator. One multiplied by seven is
seven. So that gives us nine minus seven
all over 63 is equal to two 𝑑, which is two over 63 equals two 𝑑. We can solve this equation for 𝑑
by dividing through by two. And we get that 𝑑 is equal to one
over 63.

Now that we have the value of 𝑑,
we can substitute this back into either of the equations to work out the value of
𝑎. It doesn’t matter which equation we
choose. We’ll get the answer either
way. Let’s choose the equation for the
seventh term. Substituting 𝑑 is equal to one
over 63, then gives us one-ninth is equal to 𝑎 plus six multiplied by one over
63.

Six multiplied by one over 63 is
six over 63. And we’ll need to subtract this
from both sides to calculate the value of 𝑎. 𝑎 is equal to one-ninth minus six
over 63. Once again, we choose 63 as the
lowest common denominator and we multiply both the numerator and the denominator of
our first fraction by seven. That gives us seven minus six all
over 63. And we get 𝑎 is one over 63.

So we have both a common difference
and a first term of one over 63. To find the 63rd term, we’ll
substitute each part into the formula for the 𝑛th term. That gives us one over 63 plus 63
minus one multiplied by one over 63. 63 minus one is 62. So our equation becomes one over 63
plus 62 over 63, which is 63 over 63.

Since 63 over 63 is simply one, the
63rd term in our sequence is one.