Exercise 35 Page 205

a_{n} = a_{1} + (n - 1) d

In this form,

a_{1}

is the first term of the sequence,

d

a_{n}

is the

n th

term. In the given arithmetic sequence, the first term is

a_{1} = \frac{1}{2} .

Let's observe the other terms to determine the common difference

d .

\frac{1}{2} ⟶ + \frac{1}{2} 1 ⟶ + \frac{1}{2} 1 \frac{1}{2} ⟶ + \frac{1}{2} 2 ⟶ + \frac{1}{2} \dots

The common difference is

d = \frac{1}{2} .

By substituting these two values into the explicit equation and simplifying, we can find the formula for this sequence.

a_{n} = a_{1} + (n - 1) d

$d = \frac{1}{2}$ , $a_{1} = \frac{1}{2}$

a_{n} = \frac{1}{2} + (n - 1) (\frac{1}{2})

Distribute $\frac{1}{2}$

a_{n} = \frac{1}{2} + \frac{1}{2} n - \frac{1}{2}

Subtract term

a_{n} = \frac{1}{2} n

This equation can be used to find any term in the given sequence. To find

a_{10},

the

10 th

term in the sequence, we substitute

10

for

n .

a_{n} = \frac{1}{2} n

$n = 10$

a_{10} = \frac{1}{2} (10)

$\frac{1}{b} \cdot a = \frac{a}{b}$

a_{10} = \frac{1 0}{2}

Calculate quotient

a_{10} = 5

The

10 th

term in the sequence is

5 .