A geometric sequence multiplies an initial term by a common ratio to yield the next term in the sequence. In this case, multiplying 48 by 4 yields the following term in the sequence, 192.
common ration = 1/4
12th term = 1/65536
64*1/4 = 16
16 * 1/4 = 4
4* 1/4 = 1
1 * 1/4 = 1/4
1/4*1/4 = 1/16
1/16*1/4 = 1/64
12th term = 1/65536
The 11th term in a geometric sequence is 48.
The 12th term in the sequence is 192.
The common ratio is 4.
We need to determine the 10th term of the sequence.
The general term of the geometric sequence is given by
where a is the first term and r is the common ratio.
The 11th term is given is
The 12th term is given by
Value of a:
The value of a can be determined by solving any one of the two equations.
Hence, let us solve the equation (1) to determine the value of a.
Thus, we have;
Dividing both sides by 1048576, we get;
Thus, the value of a is
Value of the 10th term:
The 10th term of the sequence can be determined by substituting the values a and the common ratio r in the general term , we get;
Thus, the 10th term of the sequence is 12.
The common ratio r is calculated as
The n th term of a geometric sequence is
where a₁ is the first term and r the common ratio
Here a₁ = 64 and r = , thus
= 64 × = 64 × = =
The terms of a geometric progression are
a, ar, ar², ar³, ...... , a
Thus the sum of the third and fourth is
ar² + ar³ = 108 → (1)
The sum of the fourth and fifth is
ar³ + a = 324 → (2)
Factorise both equations
ar²(1 + r) = 108 → (3)
ar³(1 + r) = 324 → (4)
Divide (4) by (3)
Cancelling (1 + r) , a and r, gives
r = 3
Substitute r = 3 into (3) and solve for a
9a(4) = 108
36a = 108 ( divide both sides by 36 )
a = 3
The n th term is
= a with a = 3 and r = 3, hence
= 3 = 531441
The n-th term of a geometric sequence with initial value a and common ratio r is given by .
We have and and
The explicit rule for determining the nth term of a geometric sequence is expressed as Tₙ = arⁿ⁻¹ where;
a is the first term of the geometric sequence
r is the common ratio
n is the number of terms
If a geometric sequence has a common ratio of 2 and the 12th term is −12,288, then;
T₁₂ = ar¹²⁻¹
T₁₂ = ar¹¹
Given T₁₂ = -12,288 and r = 2, we can calculate the first term a
-12,288 = a2¹¹
a = -12,288/2¹¹
a = -12,288/2048
a = -6
Since the explicit rule for determining the nth term of a geometric sequence is expressed as Tₙ = arⁿ⁻¹, then for the sequence given, the explicit rule will be;
Tₙ = -6(2)ⁿ⁻¹
Tₙ = -6 * 2ⁿ * 2⁻¹
Tₙ = -6 * 2ⁿ * 1/2
Tₙ = -3(2)ⁿ
Hence the explicit rule that describes this sequence is Tₙ = -3(2)ⁿ