The value of a term of an arithmetic progression 21, 18, 15, ... is -81. Find the term.
Understand the Problem
The question asks to find which term in the arithmetic progression (AP) 21, 18, 15, ... has a value of -81. In other words, we need to find the 'n' such that the nth term of the sequence is -81. It is a math question from a textbook or quiz.
Answer
The 35th term of the A.P. is -81.
Answer for screen readers
(C) 35th
Steps to Solve
- Identify the first term and common difference
The first term, $a$, of the arithmetic progression is 21. The common difference, $d$, is the difference between consecutive terms, which is $18 - 21 = -3$.
- Use the formula for the nth term of an AP
The $n$th term ($a_n$) of an arithmetic progression is given by the formula: $$a_n = a + (n - 1)d$$
- Substitute the given value and solve for n
We are given that $a_n = -81$. Substituting the values of $a$, $d$, and $a_n$ into the formula, we get: $$ -81 = 21 + (n - 1)(-3) $$
- Simplify and solve the equation
Simplify the equation step by step: $$ -81 = 21 - 3n + 3 $$ $$ -81 = 24 - 3n $$ $$ -81 - 24 = -3n $$ $$ -105 = -3n $$ $$ n = \frac{-105}{-3} $$ $$ n = 35 $$
- Determine the term number
Therefore, the 35th term of the arithmetic progression is -81.
(C) 35th
More Information
An arithmetic progression is a sequence of numbers such that the difference between any two consecutive terms is constant. The general form of an arithmetic progression is $a, a+d, a+2d, a+3d, ...$, where $a$ is the first term and $d$ is the common difference. The $n^{th}$ term can be obtained by $a_n = a + (n-1)d$.
Tips
A common mistake is making errors while simplifying the equation in step 4, especially with the signs. Another mistake is misidentifying the first term or calculating the common difference incorrectly.
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