Which term of the AP : 21,18,15,… is −81? Also, is any term 0? Give reason for your answer.
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Step-by-Step Solution
Step 1: Identify the first term and common difference
To find a specific term in an Arithmetic Progression (AP), we first need to identify the first term (a) and the common difference (d). The first term is given as 21. The common difference is found by subtracting any term from its succeeding term, so 18−21=−3.
Step 2: Find the term that is -81
We use the formula for the n -th term of an AP, an=a+(n−1)d. We are given an=−81, a=21, and d=−3. We substitute these values into the formula to solve for n.
Step 3: Solve for n
We simplify the equation by first subtracting 21 from both sides. Then, we divide both sides by -3 to isolate (n−1). Finally, we add 1 to find the value of n. This tells us that the 35th term of the AP is -81.
Step 4: Check if any term is 0
Now we need to determine if any term in the AP can be 0. We set an=0 in the formula for the n -th term and solve for n. If n is a positive integer, then 0 is a term in the AP.
Step 5: Solve for n when an=0
Similar to the previous calculation, we simplify the equation by subtracting 21 from both sides and then dividing by -3. The result shows that n=8. Since n is a positive integer, the 8th term of the AP is 0.
Step 6: State the reasons
The term of an AP is always represented by a positive integer. Since we found positive integer values for n in both cases, it confirms that -81 is the 35th term and 0 is the 8th term of the given Arithmetic Progression.