Question

S=1+2+2^2+2^3+2^4+...+2^100

S=1.2+2.3+3.4+4.5+...+99.100+100.101

Q=1^2+2^2+3^2+...+100^2+101^2

Answer 1

S = 1 + 2 + 2² + 2³ + 2⁴ + ... + 2¹⁰⁰

2S = 2 + 2² + 2³ + 2⁴ + ... + 2¹⁰¹

S = 2S - S

= (2 + 2² + 2³ + ... + 2¹⁰¹) - (1 + 2 + 2² + ... + 2¹⁰⁰)

= 2¹⁰¹ - 1

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S = 1.2 + 2.3 + 3.4 + ... + 99.100 + 100.101

3S = 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 99.100.(101 - 98) + 100.101.(102 - 99)

= 1.2.3 - 1.2.3 + 2

3.4 - 2.3.4 + 3.4.5 - ... - 98.99.100 + 99.100.101 - 99.100.101 + 100.101.102

= 100.101.102

S = 100 . 101 . 102 : 3

= 343400

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Q = 1² + 2² + 3² + ... + 100² + 101²

= 101.102.(2.101 + 1) : 6

= 348551