The first term of a Geometric Progression (G.P) is 6. If its common ratio is 2, find its 6th
term.
The formula to find the n-th term of a G.P. is:
aₙ = a₁ * r^(n-1)
where a₁ is the first term, r is the common ratio, and n is the term number.
Substituting the values given, we have:
a₁ = 6
r = 2
n = 6
a₆ = 6 * 2^(6-1)
a₆ = 6 * 2^5
a₆ = 6 * 32
a₆ = 192
Therefore, the 6th term of the G.P. is 192.
The sum of the first two terms of an Arithmetic Progression (A.P) is 24. The sum of the 4th and 5th terms is 36, find the common difference.
Let the first term of the arithmetic progression be a and the common difference be d.
The sum of the first two terms is 24, so we have:
a + (a+d) = 24
2a + d = 24
The sum of the 4th and 5th terms is 36, so we have:
a + 3d + a + 4d = 36
2a + 7d = 36
Now we have two equations with two variables. We can solve for d by eliminating a:
2a + d = 24
2a + 7d = 36
Subtracting the first equation from the second, we get:
6d = 12
d = 2
Therefore, the common difference is 2.
To find the 6th term of a geometric progression (G.P) with a given first term and common ratio, you can use the formula:
nth term = first term × common ratio^(n-1)
In this case, the first term is 6 and the common ratio is 2. We want to find the 6th term, so we substitute these values into the formula:
6th term = 6 × 2^(6-1)
Simplifying the exponent:
6th term = 6 × 2^5
Calculating the power of 2:
6th term = 6 × 32
Finally, multiplying:
6th term = 192
Therefore, the 6th term of the geometric progression is 192.