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

Jul 5, 2015

Answer:

The first term is ${a}_{1} = 4$

Explanation:

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:

${S}_{n} = \frac{{a}_{1} + {a}_{n}}{2} \cdot n$ (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:

${a}_{n} = {a}_{1} + \left(n - 1\right) \cdot d$ (2)

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

${S}_{n} = \frac{2 {a}_{1} + \left(n - 1\right) \cdot d}{2} \cdot n$

Now we can substitute given data and calculate ${a}_{1}$

$781 = \frac{2 {a}_{1} + 21 \cdot 3}{2} \cdot 22$

$781 = \left(2 {a}_{1} + 63\right) \cdot 11$

$71 = 2 {a}_{1} + 63$
$2 {a}_{1} = 71 - 63$
$2 {a}_{1} = 8$
${a}_{1} = 4$