How Many Terms Of The A.P. 27, 24, 21, .... Must Be Taken So That Their Sum Is 105 ? Which Term Of The A.P. Is Zero ?
Arithmetic Progression (A.P.) is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. This constant is called the common difference. In this article, we will explore how to find the number of terms in an A.P. that sum up to a given value and determine which term of the A.P. is zero.
What is Arithmetic Progression (A.P.)?
An A.P. is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. The general form of an A.P. is:
a, a + d, a + 2d, a + 3d, ...
where 'a' is the first term and 'd' is the common difference.
Example of A.P.
Let's consider an example of an A.P. with first term 'a' = 27 and common difference 'd' = -3. The A.P. will be:
27, 24, 21, 18, 15, ...
How to Find the Number of Terms in an A.P.
To find the number of terms in an A.P. that sum up to a given value, we can use the formula for the sum of an A.P.:
S_n = n/2 [2a + (n-1)d]
where S_n is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
Problem: How many terms of the A.P. 27, 24, 21, .... must be taken so that their sum is 105?
Let's use the formula for the sum of an A.P. to find the number of terms.
Given: a = 27 d = -3 S_n = 105
We need to find the value of n.
Step 1: Plug in the values into the formula
S_n = n/2 [2a + (n-1)d] 105 = n/2 [2(27) + (n-1)(-3)]
Step 2: Simplify the equation
105 = n/2 [54 + (n-1)(-3)] 105 = n/2 [54 - 3n + 3] 105 = n/2 [57 - 3n]
Step 3: Multiply both sides by 2
210 = n [57 - 3n]
Step 4: Expand the equation
210 = 57n - 3n^2
Step 5: Rearrange the equation
3n^2 - 57n + 210 = 0
Step 6: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 3, b = -57, and c = 210.
n = (57 ± √((-57)^2 - 4(3)(210))) / (2(3)) n = (57 ± √(3249 - 2520)) / 6 n = (57 ± √723) / 6
Step 7: Simplify the solutions
n = (57 ± 26.98) / 6
We have two possible solutions:
n = (57 + 26.98) / 6 n = 83.98 / 6 n = 13.99
n = (57 - 26.98) / 6 n = 30.02 / 6 n = 5.003
Since the number of terms cannot be a decimal, we round down to the nearest whole number. Therefore, the number of terms in the A.P. that sum up to 105 is 5.
Which Term of the A.P. is Zero?
To find which term of the A.P. is zero, we need to find the value of n for which the nth term is zero.
The nth term of an A.P. is given by:
a_n = a + (n-1)d
We want to find the value of n for which a_n = 0.
0 = 27 + (n-1)(-3)
Step 1: Simplify the equation
0 = 27 - 3n + 3 0 = 30 - 3n
Step 2: Rearrange the equation
3n = 30
Step 3: Solve for n
n = 30 / 3 n = 10
Therefore, the 10th term of the A.P. is zero.
Conclusion
In this article, we will answer some frequently asked questions about Arithmetic Progression (A.P.).
Q: What is the formula for the sum of an A.P.?
A: The formula for the sum of an A.P. is:
S_n = n/2 [2a + (n-1)d]
where S_n is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
Q: How do I find the number of terms in an A.P. that sum up to a given value?
A: To find the number of terms in an A.P. that sum up to a given value, you can use the formula for the sum of an A.P. and solve for n.
Q: What is the formula for the nth term of an A.P.?
A: The formula for the nth term of an A.P. is:
a_n = a + (n-1)d
where a_n is the nth term, a is the first term, d is the common difference, and n is the term number.
Q: How do I find which term of the A.P. is zero?
A: To find which term of the A.P. is zero, you can use the formula for the nth term of an A.P. and set it equal to zero.
Q: What is the difference between an A.P. and a G.P.?
A: An A.P. is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. A G.P. (Geometric Progression) is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed constant.
Q: How do I find the sum of an infinite A.P.?
A: To find the sum of an infinite A.P., you can use the formula:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Q: What is the formula for the sum of the first n terms of a G.P.?
A: The formula for the sum of the first n terms of a G.P. is:
S_n = a * (1 - r^n) / (1 - r)
where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
Q: How do I find the sum of an infinite G.P.?
A: To find the sum of an infinite G.P., you can use the formula:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Q: What is the formula for the nth term of a G.P.?
A: The formula for the nth term of a G.P. is:
a_n = a * r^(n-1)
where a_n is the nth term, a is the first term, r is the common ratio, and n is the term number.
Q: How do I find the sum of a finite A.P. or G.P.?
A: To find the sum of a finite A.P. or G.P., you can use the formulas:
For A.P.: S_n = n/2 [2a + (n-1)d]
For G.P.: S_n = a * (1 - r^n) / (1 - r)
where S_n is the sum of the first n terms, a is the first term, d is the common difference for A.P. or the common ratio for G.P., and n is the number of terms.
Conclusion
In this article, we answered some frequently asked questions about Arithmetic Progression (A.P.) and Geometric Progression (G.P.). We covered topics such as the formula for the sum of an A.P., the formula for the nth term of an A.P., and how to find the sum of an infinite A.P. or G.P. We also covered the differences between an A.P. and a G.P. and how to find the sum of a finite A.P. or G.P.