The third term of an A.P is 10 and the sixth term is 43 find: the first term of the sequence
The first term of the sequence is -23.
AAAaannndd the bot gets it wrong yet again!
10-2*11 = -12
To find the first term of the arithmetic progression (A.P), we need to use the given information that the third term is 10 and the sixth term is 43.
Let's represent the first term of the A.P as 'a'.
The common difference, denoted by 'd', is the difference between any two consecutive terms in an arithmetic progression.
In this case, the difference between the sixth term and the third term is:
43 - 10 = 33
Since the common difference is the same for all terms in an A.P, we can use this information to find the first term.
To find the first term using the common difference, we can use the formula:
a + (n-1)d = term, where n is the position of the term and term is the actual value of the term.
Using this formula, we can substitute the given values:
a + (3-1)d = 10 (for the third term)
a + (6-1)d = 43 (for the sixth term)
Substituting the value of 'a' in the second equation from the first equation, we get:
(10 + d) + (5d) = 43
Simplifying the equation:
10 + 6d = 43
Subtracting 10 from both sides:
6d = 33
Dividing both sides by 6:
d = 5.5
Now that we know the common difference is 5.5, we can substitute this back into the first equation to solve for 'a':
a + (3-1)(5.5) = 10
a + (2)(5.5) = 10
a + 11 = 10
Subtracting 11 from both sides:
a = -1
Therefore, the first term of the arithmetic progression is -1.
To find the first term of the arithmetic progression (A.P.), we need to determine the common difference. The common difference is the constant value added to each term to obtain the next term in the sequence.
Given that the third term of the A.P. is 10 and the sixth term is 43, we can use this information to find the common difference.
The formula for finding the nth term of an A.P. is:
an = a + (n - 1)d,
Where:
an is the nth term,
a is the first term,
n is the position of the term in the sequence, and
d is the common difference.
Using this formula for the third term, we have:
10 = a + (3 - 1)d.
Simplifying this equation, we get:
10 = a + 2d.
Similarly, for the sixth term, we have:
43 = a + (6 - 1)d,
43 = a + 5d.
Now we have a system of two equations with two variables (a and d):
1) 10 = a + 2d,
2) 43 = a + 5d.
To solve this system, we can use various methods, such as substitution or elimination. Let's use the elimination method.
Multiplying equation (1) by 5, we get:
50 = 5a + 10d.
Now we will subtract equation (2) from this new equation:
50 - 43 = (5a + 10d) - (a + 5d),
7 = 4a + 5d.
We now have a simplified equation:
7 = 4a + 5d.
Now, we have two equations:
1) 7 = 4a + 5d,
2) 43 = a + 5d.
To eliminate the common variable (5d), we can multiply equation (2) by 4:
4 × 43 = 4a + 4(5d),
172 = 4a + 20d.
By subtracting equation (1) from this new equation, we have:
172 - 7 = (4a + 20d) - (4a + 5d),
165 = 15d.
Finally, we can solve for d by dividing both sides of the equation by 15:
d = 165 / 15,
d = 11.
Now that we know the common difference (d = 11), we can substitute this value into either of the original equations to find the first term.
Using equation (1), we have:
10 = a + 2(11),
10 = a + 22,
Now, solve for a:
a = 10 - 22,
a = -12.
Therefore, the first term of the arithmetic progression is -12.