Question

find the number of terms in an AP given that its first and last terms are 13 and - 23 respectively and that its common difference is -2 1/4​

Answer 1

The number of terms in the arithmetic progression (AP) is 19.

Step-by-step explanation:

An arithmetic progression is defined by its first term
`(\(a\))`, common difference
`(\(d\))`, and last term
`(\(l\))`. In this case, the first term
`(\(a\))` is 13, the common difference
`(\(d\))` is -2 1/4, and the last term
`(\(l\))` is -23.

The formula to find the
`\(n\)`-th term
`(\(l\))` in an arithmetic progression is given by:

`\[ l = a + (n-1)d \]`

We are given that
`\(a = 13\), \(d = -2 (1)/(4)\), and \(l = -23\)`. Substituting these values into the formula, we get:

`\[ -23 = 13 + (n-1) \left(-2 (1)/(4)\right) \]`

Now, solve for
`\(n\)`:

`\[ -23 = 13 - 2n + (9)/(4) \]`

`\[ -36 = -2n + (9)/(4) \]`

`\[ -(147)/(4) = -2n \]`

Divide both sides by
`\(-2\)` to solve for
`\(n\)`:

`\[ (147)/(8) = n \]`

So,
`\(n = 18.375\)`. However, the number of terms
`(\(n\))`must be a whole number since it represents the count of terms in the sequence. Therefore, we round up to the nearest whole number, and the number of terms in the arithmetic progression is 19.

In conclusion, the AP has 19 terms. This means that the sequence starts at 13, decreases by -2 1/4 for 18 times, and reaches -23 at the end of the progression.

Answer 2

There are 17 terms in the sequence

Step-by-step explanation:

Arithmetic Sequence

An arithmetic sequence is a list of numbers with a definite pattern by which each term is calculated by adding or subtracting a constant number called common difference to the previous term. If n is the number of the term, then:

`a_n=a_1+(n-1)r`

Where an is the nth term, a1 is the first term, and r is the common difference.

In the problem at hand, we are given the first term a1=13, the last term an=-23, and the common difference r=-2 1/4. Let's solve the equation for n:

`\displaystyle n=1+(a_n-a_1)/(r)`

We need to express r as an improper or proper fraction:

`\displaystyle r=-2(1)/(4)=-2-(1)/(4)=-(9)/(4)`

Substituting:

`\displaystyle n=1+(-23-13)/(-(9)/(4))`

`\displaystyle n=1+(-36)/(-(9)/(4))`

`\displaystyle n=1+36*(4)/(9)=17`

n=17

There are 17 terms in the sequence