the 20th sum of an arithmetic progression is -7 and the sum of the first 20 terms is 620. find the 1st and 30th terms

i am looking at using S20 = a(1-rn)/l-r = 620. but i am reallys tuck and think i am very wrong. please help.

First of all, you are using the formula for a geometric, yours is an arithmetic

Secondly, the sentence
"the 20th sum of an arithmetic progression is -7 and the sum of the first 20 terms is 620" makes no sense to me

what does "the 20th sum" mean ?

To find the first and 30th terms of the arithmetic progression, we can use the formula for the sum of an arithmetic progression.

First, let's find the common difference (d). We know that the 20th term is -7, so we can use the formula for the nth term, Tn = a + (n-1)d, where Tn is the nth term, a is the first term, n is the number of terms, and d is the common difference.

So, we have -7 = a + (20-1)d.

Next, we need to find the sum of the first 20 terms (S20), which is given as 620. The formula for the sum of an arithmetic progression is S = (n/2)(2a + (n-1)d), where S is the sum, n is the number of terms, a is the first term, and d is the common difference.

Substituting the values, we have 620 = (20/2)(2a + (20-1)d).

Now we have two equations:

-7 = a + 19d -- Equation (1)
620 = 10(2a + 19d) -- Equation (2)

To solve this system of equations, we can solve Equation (1) to express "a" in terms of "d", and then substitute it into Equation (2) to solve for "d".

From Equation (1), we get a = -7 - 19d.

Substituting this value into Equation (2), we have:

620 = 10[2(-7 - 19d) + 19d]
620 = 10[-14 - 38d + 19d]
620 = 10(-14 - 19d)
620 = -140 - 190d
190d = -140 - 620
190d = -760
d = -760/190
d = -4

Now that we have the value of "d" as -4, we can substitute it back into Equation (1) to find the value of "a".

From Equation (1), we have -7 = a + 19(-4).
-7 = a - 76
a = -7 + 76
a = 69

Therefore, the first term (a) is 69 and the common difference (d) is -4.

To find the 30th term, we can use the formula for the nth term:

T30 = a + (30-1)d
T30 = 69 + 29(-4)
T30 = 69 - 116
T30 = -47

So, the 30th term is -47.

In conclusion, the first term of the progression is 69 and the 30th term is -47.