Which term of the AP: 21, 18, 15, . . . is – 81? Also, is any term 0?
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Step-by-Step Solution
Step 1: Identify the first term and common difference
To solve this arithmetic progression (AP) problem, 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. Here, we subtract the first term from the second term, 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 looking for the term n when an=−81. Substitute the known values of a, d, and an into the formula to form an equation.
Step 3: Solve for n
Now we solve the equation for n. First, subtract 21 from both sides. Then, divide both sides by −3. Finally, add 1 to find the value of n. This value represents the position of −81 in the arithmetic progression.
Step 4: Check if any term is 0
To determine if 0 is a term in the AP, we set an=0 in the formula for the n -th term, an=a+(n−1)d. Substitute the first term a=21 and the common difference d=−3 into the equation.
Step 5: Solve for n when an=0
Solve this equation for n. Subtract 21 from both sides, then divide by −3. Add 1 to the result to find n. Since n is a positive integer, 0 is indeed a term in the AP.