To find the 6th term of the geometric progression (g.p.), we need to determine the common ratio (r) first.
Using the formula for nth term of a g.p., we can write the equation for the 3rd term as:
a × r² = 54 ...(1)
Similarly, for the 9th term, we have:
a × r⁸ = 39336 ...(2)
Dividing equation (2) by equation (1), we get:
r⁸/r² = 39336/54
r⁶ = 729
r = 3
Now we know the common ratio (r = 3).
To find the 6th term, we can use the formula for nth term of a g.p.:
a × r⁵ = a × 3⁵ = a × 243
We can find the value of 'a' by substituting the value of the 3rd term (54):
54 × r² = a × 3²
54 × 9 = a × 9
a = 6
Thus, the 6th term of the g.p. is:
6 × 243 = 1458
To find the sum of the 4th term, we can again use the formula for nth term of a g.p.:
a × r³ = a × 3³ = a × 27
Summing the 4th term involves adding the terms from the 1st term to the 4th term.
The sum of n terms in a g.p. can be calculated using the formula:
Sum(n) = a × (rⁿ - 1) / (r - 1)
Substituting n = 4, a = 6, and r = 3, we can find the sum of the first 4 terms:
Sum(4) = 6 × (3⁴ - 1) / (3 - 1)
= 6 × (81 - 1) / 2
= 6 × 80 / 2
= 6 × 40
= 240
The sum of the 4th and 7th terms can be found by summing the terms from 1st to 4th and from 1st to 7th.
Again using the formula for the sum of n terms, we find:
Sum(7) = a × (r⁷ - 1) / (r - 1)
Substituting n = 7, a = 6, and r = 3, we can calculate the sum of the first 7 terms:
Sum(7) = 6 × (3⁷ - 1) / (3 - 1)
= 6 × (2187 - 1) / 2
= 6 × 2186 / 2
= 6 × 1093
= 6558
The product of the 2nd and 5th terms can be calculated as:
a × r × r⁴ = a × r⁵
Substituting a = 6 and r = 3, we get:
6 × 3⁵ = 6 × 243
= 1458
Therefore, the 6th term is 1458, the sum of the 4th term is 240, the sum of the 4th and 7th terms is 6558, and the product of the 2nd and 5th terms is 1458.