An arithmetic series has first term 5 and a common difference 7

a) write down the 4th term and the 23rd term

b) find the sum of the first 38 terms of this series.

Tn = 5 + 7(n-1) = 7n-2

S38 = 38/2 (2*5 + 37*7)

To find the 4th term of the arithmetic series, we use the formula:

nth term = first term + (n - 1) * common difference

a) The 4th term:
nth term = 5 + (4 - 1) * 7
= 5 + 3 * 7
= 5 + 21
= 26

Therefore, the 4th term is 26.

To find the 23rd term of the arithmetic series:
nth term = 5 + (23 - 1) * 7
= 5 + 22 * 7
= 5 + 154
= 159

Therefore, the 23rd term is 159.

b) To find the sum of the first 38 terms of the arithmetic series, we use the formula for the sum of an arithmetic series:

sum = (n/2) * (first term + last term)

In this case, n = 38.

first term = 5
last term = 5 + (38 - 1) * 7
= 5 + 37 * 7
= 5 + 259
= 264

sum = (38/2) * (5 + 264)
= 19 * 269
= 5111

Therefore, the sum of the first 38 terms of this arithmetic series is 5111.

To find the terms of an arithmetic series, we can use the formula:

nth term = first term + (n - 1) * common difference

where "n" is the position of the term in the series.

a) The 4th term can be found by substituting n = 4 into the formula:

4th term = 5 + (4 - 1) * 7
= 5 + 3 * 7
= 5 + 21
= 26

Therefore, the 4th term of the arithmetic series is 26.

To find the 23rd term, we substitute n = 23 into the formula:

23rd term = 5 + (23 - 1) * 7
= 5 + 22 * 7
= 5 + 154
= 159

Therefore, the 23rd term of the arithmetic series is 159.

b) To find the sum of the first 38 terms of the series, we use another formula:

Sum of the first n terms = (n/2) * (2a + (n - 1)d)

where "a" is the first term, "d" is the common difference, and "n" is the number of terms.

Substituting the given values, we have:

Sum of the first 38 terms = (38/2) * (2 * 5 + (38 - 1) * 7)
= 19 * (10 + 37 * 7)
= 19 * (10 + 259)
= 19 * 269
= 5111

Therefore, the sum of the first 38 terms of the arithmetic series is 5111.