if the 7th term of an a.p is twice the third term and the sum of the first four terms is-16 and the last and the last term is 72, find the number of the terms
Let an be the nth term of arithmetic progression a. Can you then start to write out your ap using symbols?
not so.
I'll show you how to find the answer.
Using the formula for the nth term of an AP, you know that
a+6d = 2(a+2d)
4/2 (2a+3d) = 16
Now you can find a and d, so you just need to know how many terms there are.
If there are k terms, then
a+(k-1)d = 72
so using a and d found in part 1, solve for k.
To find the number of terms in an arithmetic progression (a.p.), we need to first determine the common difference (d) of the sequence. Once we have the common difference, we can calculate the number of terms using the given information.
Let's first find the common difference (d):
Given that the 7th term (a₇) is twice the third term (a₃), we can set up the following equation:
a₇ = 2a₃
Using the formula for the nth term of an arithmetic progression (aₙ = a₁ + (n-1)d), we can express a₇ and a₃ in terms of the first term (a₁) and the common difference (d):
a₁ + 6d = 2(a₁ + 2d)
Simplifying the equation further:
a₁ + 6d = 2a₁ + 4d
-3d = a₁
Next, we know that the sum of the first four terms is -16. We can use the formula for the sum of an arithmetic progression (Sₙ = (n/2)(2a₁ + (n-1)d)) to express this relation:
S₄ = (4/2)(2a₁ + (4-1)d) = 2(2a₁ + 3d)
Given that S₄ = -16, we can substitute this information into the equation:
2(2a₁ + 3d) = -16
4a₁ + 6d = -16
2a₁ + 3d = -8 (dividing both sides of the equation by 2)
Now, let's consider the last term of the arithmetic progression, which is 72. Using the formula for the nth term, we can express this relation as:
aₙ = a₁ + (n-1)d
aₙ = a₁ + (N-1)d = 72 (N represents the number of terms)
Since we want to find the number of terms (N), we need to express aₙ in terms of a₁ and d. Recall that we found a₁ in terms of d:
-3d = a₁
Substituting this into the equation for aₙ:
-3d + (N-1)d = 72
Simplifying this equation, we get:
-2d + Nd = 72
(N - 2)d = 72
(N - 2)d = 2³ × 3² (72 can be factored as 2³ × 3²)
The only possible values for d are 1, 2, 4, 8, 3, 6, 9, 12, 18, 24 since the common difference should be an integer.
If d = 1, then N - 2 = 2³ × 3² = 72, which is not possible as N - 2 should be an integer factor.
If d = 2, then N - 2 = 2² × 3² = 36, which is possible. Solving for N:
N - 2 = 36
N = 36 + 2
N = 38
If d = 4, then N - 2 = 2 × 3² = 18, which is not possible as N - 2 should be an integer factor.
If d = 8, then N - 2 = 3² = 9, which is not possible as N - 2 should be an integer factor.
If d = 3, then N - 2 = 2³ × 3 = 24, which is not possible as N - 2 should be an integer factor.
If d = 6, then N - 2 = 2² × 3 = 12, which is not possible as N - 2 should be an integer factor.
If d = 9, then N - 2 = 3² = 9, which is not possible as N - 2 should be an integer factor.
If d = 12, then N - 2 = 2 × 3 = 6, which is not possible as N - 2 should be an integer factor.
If d = 18, then N - 2 = 2² = 4, which is not possible as N - 2 should be an integer factor.
If d = 24, then N - 2 = 3 = 3, which is not possible as N - 2 should be an integer factor.
Therefore, the only possible value for the common difference (d) is 2, and the number of terms (N) is 38.
Hence, the solution to the problem is N = 38.