For these arithmetic progressions, indicate the value of the difference d, write down the formula for the nth term (an), find S4.

| August 22, 2021 | education

The task consists of two parts, in each of which, using the data of the arithmetic progression, you need to find the value of the difference d, write down the formula for the n-th member of an and calculate the sum of the first four terms, that is, calculate the value of S4.

A) Consider a sequence of numbers 18, 23, 28, …. First of all, note that since a2 – a1 = 23 – 18 = 5 and a3 – a2 = 28 – 23 = 5, then this sequence is an arithmetic progression with a1 = 18 and the difference d = 5. Using the formula an = a1 + d * (n – 1), we have an = 18 + 5 * (n – 1) = 18 + 5 * n – 5 = 23 + 5 * n. According to the formula Sn = (2 * a1 + d * (n – 1)) * n / 2, we calculate S4 = (2 * 18 + 5 * (4 – 1)) * 4/2 = (36 + 15) * 4 / 2 = 102.

Answers: d = 5, an = 23 + 5 * n and S4 = 102.

B) Consider a sequence of numbers 18, 15, 12, …. First of all, note that since a2 – a1 = 15 – 18 = -3 and a3 – a2 = 12 – 15 = -3, this sequence is an arithmetic progressions with a1 = 18 and the difference d = -3. Using the formula an = a1 + d * (n – 1), we have an = 18 + (-3) * (n – 1) = 18 – 3 * n + 3 = 21 – 3 * n. According to the formula Sn = (2 * a1 + d * (n – 1)) * n / 2, we calculate S4 = (2 * 18 + (-3) * (4 – 1)) * 4/2 = (36 – 9 ) * 4/2 = 54.

Answers: d = -3, an = 21 – 3 * n and S4 = 54.



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