How do you find the first term of the arithmetic sequence given Sn = 781, d = 3, n = 22?

Question

How do you find the first term of the arithmetic sequence  given Sn = 781, d = 3, n = 22?

Cora Alexander · Accepted Answer

The first term is #a_1=4#

In this sequence we know the sum of 22 terms, the difference and we have to calculate the first term. We will use the formula for the sum of n terms:

#S_n=(a_1+a_n)/2*n# (1)

In this formula we have all data except #a_1#, which we want to find, and #a_n#, but the last term can be calculated using:

#a_n=a_1+(n-1)*d# (2)

When we substitute (2) to (1) we get:

#S_n=(2a_1+(n-1)*d)/2*n#

Now we can substitute given data and calculate #a_1#

#781=(2a_1+21*3)/2*22#

#781=(2a_1+63)*11#

#71=2a_1+63# #2a_1=71-63# #2a_1=8# #a_1=4#

Xavier Alonso · Answer

undefined To find the first term $a$ of the arithmetic sequence given $S_n = 781$, $d = 3$, and $n = 22$, we use the formula for the sum of the first $n$ terms of an arithmetic sequence:

$$ S_n = \frac{n}{2}[2a + (n - 1)d] $$

Given $S_n = 781$, $d = 3$, and $n = 22$, we can plug these values into the formula and solve for $a$:

$$ 781 = \frac{22}{2}[2a + (22 - 1) \times 3] $$
$$ 781 = 11[2a + 21 \times 3] $$
$$ 781 = 11[2a + 63] $$
$$ 781 = 22a + 693 $$
$$ 22a = 781 - 693 $$
$$ 22a = 88 $$
$$ a = \frac{88}{22} $$
$$ a = 4 $$

Therefore, the first term $a$ of the arithmetic sequence is 4.