Progressions

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Step 1: Find the difference between consecutive terms. 8500 - 8000 = 500 9000 - 8500 = 500 Step 2: Since the difference is the same each time, the common difference is 500. Answer: d = 500…

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Step 1: Find the common difference. 43 - 45 = -2 Step 2: Add -2 to the last term each time. 39 + (-2) = 37 37 + (-2) = 35 Answer: 37, 35…

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Step 1: Check consecutive differences. 1 - 1 = 0 2 - 1 = 1 3 - 2 = 1 Step 2: The differences are not equal. Answer: No, it is not an AP.

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Use the formula: a_n = a + (n - 1)d Step 1: Substitute a = 6 and d = 3. a_n = 6 + (n - 1)3 Step 2: Simplify. a_n = 6 + 3n - 3 = 3n + 3 Answer: a_n = 6 + (n - 1)3 = 3n + 3…

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Step 1: Identify the first term and common difference. a = 5, d = 1 - 5 = -4 Step 2: Use the nth term formula. a_n = a + (n - 1)d Step 3: Substitute n = 10. a_10 = 5 + (10 - 1)(-4) a_10 = 5 + 9(-4) a_…

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Step 1: Find a and d. a = 21, d = 18 - 21 = -3 Step 2: Use a_n = a + (n - 1)d. -81 = 21 + (n - 1)(-3) Step 3: Solve. -81 = 21 - 3n + 3 -81 = 24 - 3n -105 = -3n n = 35 Answer: -81 is the 35th term.

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Step 1: Find the first term and common difference. a = 5, d = 11 - 5 = 6 Step 2: Use a_n = a + (n - 1)d. 301 = 5 + (n - 1)6 Step 3: Solve. 301 = 5 + 6n - 6 301 = 6n - 1 302 = 6n n = 151/3 Step 4: n mu…

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Step 1: Write the two-digit multiples of 3. 12, 15, 18, ..., 99 Step 2: This is an AP with a = 12 and d = 3. Step 3: Use a_n = a + (n - 1)d. 99 = 12 + (n - 1)3 99 = 12 + 3n - 3 99 = 3n + 9 90 = 3n n =…