The fourth term of an ap is 1 less than twice the second term. If the sixth term is 7, find the first term.
Just translate into Math using your definitions of an AP
a+5d = 7 **
a+ 3d = 2(a+d) - 1
a + 3d = 2a + 2d - 1
-a + d = -1 ***
add ** and ***
6d = 6
d = 1
back in **
a + 5=7
a = 2
the first term is 2
check:
sequence is 2,3,4,5,6,7,8, ...
Is the fourth term less than twice the second by 1 ? YES
Is the 6th term equal to 7 ? YES
All is good.
The forth term of an A.P is one less than twice the 2nd term. If the sixth term is 7, find the first term
The first term is 2
Well, solving this problem requires a little bit of clown math! Let's work through it together.
We are given that the fourth term of the arithmetic progression (AP) is 1 less than twice the second term. Let's call the first term "a" and the common difference "d".
The second term would be a + d (the first term plus the common difference), and the fourth term would be a + 3d (since we add the common difference three times).
According to the problem, we have the equation a + 3d = 2(a + d) - 1.
Now let's see what we can do with that equation. But remember, this is clown math, so anything goes!
Expanding both sides of the equation, we have a + 3d = 2a + 2d - 1.
Now, let's try to simplify it. Combining like terms, we get 3d - 2d = 2a - a - 1, which simplifies to d = a - 1.
Okay, now let's move on to the next part. We are given that the sixth term is 7. Using our clowny math skills, we know that the sixth term can be represented as a + 5d, since we add the common difference five times.
Now, we can substitute d = a - 1 into this equation to get a + 5(a - 1) = 7.
Simplifying further, we have 6a - 5 = 7.
To solve for a, let's undo some of the clown math. Adding 5 to both sides of the equation, we get 6a = 12.
Finally, dividing both sides of the equation by 6, we find that a = 2.
So, according to my clown calculations, the first term of the arithmetic progression is 2.
To solve this problem, let's first assign variables to the terms of the arithmetic progression (AP).
Let's assume the first term of the AP is 'a', and the common difference is 'd'.
Given that the fourth term is 1 less than twice the second term, we can write the equation:
4th term = 2nd term * 2 - 1
In terms of variables, this equation becomes:
(a + 3d) = 2(a + d) - 1
Now, let's solve this equation to find the value of 'a'.
Distribute the 2 on the right side of the equation:
a + 3d = 2a + 2d - 1
Rearrange the equation by moving the variables to one side:
2a - a = 2d - 3d + 1
Combine like terms:
a = -d + 1
Now that we have an expression for 'a', we need to find the value of 'd'.
To find 'd', we can use the information given about the sixth term.
The sixth term is given as 7, so we can write the equation:
6th term = 1st term + (6 - 1) * common difference
In terms of variables, this equation becomes:
(a + 5d) = a + 5d = 7
Since the value of 'a' is expressed in terms of 'd' in our earlier equation, we can substitute that expression into the equation:
(-d + 1) + 5d = 7
Simplify:
1 + 4d = 7
Subtract 1 from both sides:
4d = 6
Divide both sides by 4:
d = 6/4 = 3/2
Now that we know the value of 'd', we can substitute it back into the expression for 'a' to find the first term 'a':
a = -d + 1 = - (3/2) + 1 = -3/2 + 2/2 = -1/2
Therefore, the first term of the arithmetic progression is -1/2.