The sum of an arithmeit cprogression consisting of 20 positive integer terms with positive common difference is equal to 2020. ( a) how many progressions are possible (b) what are the smallest and largest possble values of the first term?
1 answer:
Answer:
a) possible progressions are 5
b) the smallest and largest possible values of the first term are 16 and 82
Step-by-step explanation:
<u>Sum of terms:</u>
Sₙ = n/2(a₁ + aₙ) = n/2(2a₁ + (n-1)d) S₂₀ = 20/2(2a₁ + 19d) = 10(2a₁ + 19d) 2020 = 10(2a₁ + 19d) 202 = 2a₁ + 19d <u>In order a₁ to be an integer, d must be even number, so d = 2k</u>
202 = 2a₁ + 38k 101 = a₁ + 19k <u>Possible values of k= 1,2,3,4,5</u>
k = 1 ⇒ a₁ = 101 - 19 = 82 k = 2 ⇒ a₁ = 101 - 38 = 63 k = 3 ⇒ a₁ = 101 - 57 = 44 k = 4 ⇒ a₁ = 101 - 76 = 25 k = 5 ⇒ a₁ = 101 - 95 = 16 <u>As per above, </u>
a) possible progressions are 5 b) the smallest and largest possible values of the first term are 16 and 82
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