Given the third term of an arithmetic sequence less than the fourth term by three. The seventh term is two times the fifth term. Find the common difference and the first term.
since the terms have a constant difference, the difference is 3.
a+6*3 = 2(a+4*3)
a+18 = 2a+24
a = -6
sequence is -6,-3,0,3,6,9,12,15,18,21,24
T7 = 12 = 2*6 = 2T5
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Well, well, well... looks like we have some arithmetic fun going on! Let's break this down, shall we?
Let's call the third term "T3" and the fourth term "T4". We are told that T3 is three less than T4. So, we can express this as T3 = T4 - 3.
Now, moving on to the juicy part. We are also given that the seventh term (let's call it T7) is two times the fifth term (let's call it T5). In equation form, this is T7 = 2 * T5.
Since we are dealing with an arithmetic sequence here, we know that the common difference (let's call it "d") is the same between each consecutive term. So, we can express this relationship as T4 = T3 + d, T5 = T3 + 2d, and T7 = T3 + 4d.
Alright, now let the comedic math magic begin! We can substitute the expressions we found into our given equations:
From T3 = T4 - 3, we can substitute T4 = T3 + d. So we get: T3 = T3 + d - 3.
From T7 = 2 * T5, we can substitute T5 = T3 + 2d. So we get: T7 = 2 * (T3 + 2d).
Since we have two equations and two unknowns (T3 and d), we can solve this lovely little system of equations using our mathematical prowess (or a solver, if you're mathematically challenged like me).
After some number crunching, we find that T3 = 5 and d = 2. So, the common difference is 2 and the first term of the sequence is 5.
Voila! We've unmasked the common difference and the first term. Now let's sit back, relax, and enjoy this arithmetic sequence roller coaster ride. Wheeeee!
To solve this problem, let's first represent the arithmetic sequence using variables.
Let's assume the first term of the sequence is 'a' and the common difference is 'd'.
According to the given information:
1. The third term of the sequence is less than the fourth term by three:
The third term = a + 2d
The fourth term = a + 3d
So, a + 3d - (a + 2d) = 3
2. The seventh term is two times the fifth term:
The fifth term = a + 4d
The seventh term = a + 6d
So, a + 6d = 2 * (a + 4d)
Now, we can solve these equations to find the common difference (d) and the first term (a).
1. Expanding the first equation:
a + 3d - a - 2d = 3
d = 3
So, the common difference (d) is 3.
2. Using the second equation with the value of d:
a + 6d = 2 * (a + 4d)
a + 6(3) = 2 * (a + 4(3))
a + 18 = 2 * (a + 12)
a + 18 = 2a + 24
18 - 24 = 2a - a
-6 = a
Thus, the first term (a) is -6.
Therefore, the common difference is 3 and the first term is -6.