if p+qth term of GS is m and p-qth term is n find pth and qth term

T(p+q) = ar^(p+q-1) = m

T(p-q) = ar^(p-q-1) = n

multiplying, r^(2p-2) = mn
dividing, r^(2q) = m/n

you should be able to use that to find
T(p) and T(q)

To find the pth and qth terms of a geometric sequence (GS) given the (p+q)th term and the (p-q)th term, let's follow these steps:

Step 1: Write down the formula for the general term of a geometric sequence
The general formula for the nth term of a geometric sequence is given by:
an = a * r^(n-1)
where:
- an represents the nth term,
- a is the first term of the sequence, and
- r is the common ratio of the sequence.

Step 2: Express the (p+q)th and (p-q)th terms in terms of a, r, p, and q
According to the given problem, the (p+q)th term is represented by m, and the (p-q)th term is represented by n. Therefore, we have:
a * r^(p+q-1) = m ...(Equation 1)
a * r^(p-q-1) = n ...(Equation 2)

Step 3: Solve Equations 1 and 2 simultaneously
We can solve the above equations simultaneously to find the values of a and r.

Divide Equation 1 by Equation 2 to get rid of 'a':
(r^(p+q-1))/(r^(p-q-1)) = m/n

Apply the rule of exponents, which states that when you divide two exponential expressions with the same base, subtract the exponents:
r^(2q) = (m/n)

Taking the square root of both sides, we get:
r^q = sqrt(m/n)

Now, substitute this value of r in Equation 2:
a * sqrt(m/n)^(p-q-1) = n

We can rearrange this equation to solve for a:
a = n * sqrt(n/m)^(q-p+1)

Step 4: Substitute the values of a and r in the general term formula
Now that we have obtained the values of a and r, substitute them back into the general term formula:
The pth term (ap) = a * r^(p-1)
The qth term (aq) = a * r^(q-1)

Therefore, the pth term is:
ap = (n * sqrt(n/m)^(q-p+1)) * sqrt(m/n)^(p-1)

And the qth term is:
aq = (n * sqrt(n/m)^(q-p+1)) * sqrt(m/n)^(q-1)

By following these steps, you can calculate the pth and qth terms of the given geometric sequence in terms of the (p+q)th and (p-q)th terms.

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