In an arithmetic progression the 22nd term is four times the 5th term while the 12th term is 12th more than 8th term.find the first term and the common difference
Let's assume that the first term of the arithmetic progression is represented by 'a' and the common difference is represented by 'd'.
We are given two conditions:
1) The 22nd term is four times the 5th term.
Using the formula for the nth term of an arithmetic progression: an = a + (n - 1) * d, we can write this condition as:
a + (22 - 1) * d = 4 * (a + (5 - 1) * d)
Simplifying the equation, we get:
a + 21d = 4a + 4d
Rearranging the equation, we get:
3a = 17d .........(equation 1)
2) The 12th term is 12 more than the 8th term.
Using the same formula, we can write this condition as:
a + (12 - 1) * d = (a + (8 - 1) * d) + 12
Simplifying the equation, we get:
a + 11d = a + 7d + 12
Rearranging the equation, we get:
4d = 12
d = 3 .........(equation 2)
Now, substitute equation 2 into equation 1 to solve for 'a':
3a = 17 * 3
3a = 51
a = 17
Therefore, the first term of the arithmetic progression is 17 and the common difference is 3.
If the first term of an AP is 4 and common difference is -2, what is its 2nd term If the first term of an AP is 4 and common difference is -2, what is its 2nd term
Well, it seems like this arithmetic progression has some interesting quirks! Let's have some fun figuring it out.
Let's start by defining the first term as 'a' and the common difference as 'd'.
Based on the given information, we can form two equations:
1) The 22nd term is four times the 5th term:
a + 21d = 4(a + 4d)
2) The 12th term is 12 more than the 8th term:
a + 11d = (a + 7d) + 12
Now, let me put on my imaginary hat and calculate the values for you.
Solving the first equation, we simplify it to:
a + 21d = 4a + 16d,
3a = 5d.
Now, let's solve the second equation:
a + 11d = a + 7d + 12,
4d = 12,
d = 3.
Using this value, we can find 'a' from our first equation:
3a = 5d,
3a = 5 * 3,
3a = 15,
a = 5.
So, the first term of this arithmetic progression is 5, and the common difference is 3.
Now, wasn't that a Math Circus of a problem? Enjoy your arithmetic adventure!
To find the first term and common difference of an arithmetic progression, we can use the given information about the terms.
Let's first define the arithmetic progression:
The first term of the arithmetic progression is denoted by "a," and the common difference is denoted by "d."
Given information:
- The 22nd term is four times the 5th term: a + 21d = 4(a + 4d)
- The 12th term is 12 more than the 8th term: a + 11d = (a + 7d) + 12
Now we have a system of two equations with two unknowns:
Eq1: a + 21d = 4(a + 4d)
Eq2: a + 11d = (a + 7d) + 12
Simplifying Eq1:
a + 21d = 4a + 16d
3a = 5d …………….…(3a - 5d = 0) (Eq1 simplified)
Simplifying Eq2:
a + 11d = a + 7d + 12
4d = 12
d = 3
Substituting the value of "d" into Eq1 (3a - 5d = 0):
3a - 5(3) = 0
3a = 15
a = 5
Therefore, the first term (a) is 5, and the common difference (d) is 3 in the arithmetic progression.
a+21d = 4(a+4d)
since a12 = a8+12, 4d=12
Now you can finish it off.