4.2 Q-3 | Sequence And Series | Mathematics 11

Exercise 4.2 — Question 3

This question involves applying the arithmetic sequence formula to find unknown terms, the common difference, or the number of terms.

The $n^{t h}$ term of an arithmetic sequence is:

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

where:

The common difference is found by subtracting any term from the next:

$d = a_{2} - a_{1} = a_{3} - a_{2} = a_{n} - a_{n - 1}$

If $a_{1}$ , $d$ , and $a_{n}$ are known, solve for $n$ :

$n = \frac{a _{n} - a _{1}}{d} + 1$

The sum of the first $n$ terms:

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

Find the 15th term of the arithmetic sequence: $3, 7, 11, 15, \dots$

Step 1: Identify $a_{1} = 3$ and $d = 7 - 3 = 4$ .

Step 2: Apply the formula: $a_{15} = 3 + (15 - 1) (4) = 3 + 56 = 59$

Find the number of terms in the sequence $5, 8, 11, \dots, 50$ :

Step 1: $a_{1} = 5$ , $d = 3$ , $a_{n} = 50$ .

Step 2: $n = \frac{50 - 5}{3} + 1 = \frac{45}{3} + 1 = 15 + 1 = 16$