Find the sum of the series 1.2 + 2.22 + 3.23 + … + 100. 2100. (a) 1

1.	Find the sum of the series 1.2 + 2.22 + 3.23 + … + 100. 2100. (a) 100.2101 + 2 (b) 99.2100 + 2 (c) 99.2101 + 2 (d) None of these
Answer» Correct option (c) 99.2¹⁰¹ + 2 Explanation: Solve this based on pattern of the options. The given series has 100 terms. For n = 100, the options can be converted as Option (a) = n × 2^{(n + 1)} + 2. This means that for 1 term, the sum should be 1 × 2² + 2 = 6. But we can see that for 1 term, the series has a sum of only 1 × 2 = 2. Hence, this option can be rejected. Option (c) satisfies the conditions. Option (b) = (n − 1) × 2ⁿ + 2. For 1 term, this gives us a value of 2. For 2 terms, this gives us a value of 6, which does not match the actual value in the question.Hence, this option can be rejected.

Find the sum of the series 1.2 + 2.22 + 3.23 + … + 100. 2100. (a) 100.2101 + 2 (b) 99.2100 + 2 (c) 99.2101 + 2 (d) None of these

Answer»

Correct option (c) 99.2¹⁰¹ + 2

Explanation:

Solve this based on pattern of the options. The given series has 100 terms. For n = 100, the options can be converted as

Option (a) = n × 2^{(n + 1)} + 2. This means that for 1 term, the sum should be 1 × 2² + 2 = 6. But we can see that for 1 term, the series has a sum of only 1 × 2 = 2. Hence, this option can be rejected.

Option (c) satisfies the conditions.

Option (b) = (n − 1) × 2ⁿ + 2. For 1 term, this gives us a value of 2. For 2 terms, this gives us a value of 6, which does not match the actual value in the question.Hence, this option can be rejected.

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