if a3 = 20 and a6=41 for an arithmetic sequence, find a1
a(6)=a(1)+(6-1)d
a(3)=a(1)+(3-1)d
Subtract:
a(6)-a(3)=3d
41-20=3d
d=7
from which
a(3)=a(1)+(3-1)7
a(1)=20-14=6
To find the first term, we need to determine the common difference of the arithmetic sequence. Then, we can use the formula for the nth term of an arithmetic sequence.
Step 1: Determine the common difference (d):
To find the common difference, we can subtract any two consecutive terms:
d = a6 - a3
d = 41 - 20
d = 21
Step 2: Find the first term (a1):
To find the first term, we can use the formula:
a1 = a6 - (6-1) * d
Plugging in the values we have:
a1 = 41 - (6-1) * 21
a1 = 41 - 5 * 21
a1 = 41 - 105
a1 = -64
Therefore, the first term (a1) of the arithmetic sequence is -64.
To find the value of a1 in an arithmetic sequence, we need to find the common difference (d) first. Once we have the common difference, we can use the formula for the nth term of an arithmetic sequence to find a1.
In this case, we are given that a3 = 20 and a6 = 41.
Step 1: Find the common difference (d)
To find the common difference, we need to find the difference between any two consecutive terms in the sequence. We can use a3 and a6 for this.
The difference between a3 and a6 is:
d = a6 - a3
d = 41 - 20
d = 21
So the common difference is 21.
Step 2: Use the formula for the nth term of an arithmetic sequence to find a1
The formula for the nth term of an arithmetic sequence is given by:
an = a1 + (n - 1) * d
We know that a3 = 20 and n = 3. Substituting these values into the formula, we can solve for a1.
a3 = a1 + (3 - 1) * d
20 = a1 + 2 * 21
20 = a1 + 42
Subtracting 42 from both sides:
a1 = 20 - 42
a1 = -22
Therefore, a1 = -22.