Question

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?

Answer 1

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:

Sum of terms:

• 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

In order a₁ to be an integer, d must be even number, so d = 2k

• 202 = 2a₁ + 38k
• 101 = a₁ + 19k

Possible values of k= 1,2,3,4,5

• 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

As per above,

• a) possible progressions are 5
• b) the smallest and largest possible values of the first term are 16 and 82