40 the first term of an arithmetic sequence is -12. the common difference of the sequence is 7. what is the sum of the first 30 terms of the sequence?

 2685

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Explanation:

When we want to add up the first n terms of an arithmetic sequence, the formula to use is

S = (n/2)*( 2*a+d(n-1) )

where

S = sum of the first n terms

n = number of terms

a = first term

d = common difference

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In this case,

S = unknown

n = 30

a = -12

d = 7

which means,

S = (n/2)*( 2*a + d*(n-1) )

S = (30/2)*( 2*(-12) + 7*(30-1) )

S = 15*(-24 + 7*29)

S = 15*(-24 + 203)

S = 15*(179)

S = 2685

This answer is confirmed using a spreadsheet. Basically I had the spreadsheet generate 30 terms based on a pattern I gave it of the first two terms. Then I used the "SUM" function to add up all 30 terms quickly getting 2685.

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Side note:

Another formula we could use is

S = (n/2)*(a_1 + a_n)

where

a_1 = first term

a_n = nth term, when n = 30 this is the 30th term

The a_n part is equal to a_n = a_1 + d(n-1), and when you add this to the a_1 already in the S formula, that accounts for the 2*a_1 back in the first formula mentioned at the top of the page.

sum of the first 30 terms is 2685

Step-by-step explanation:

The formula for determining the sum of n terms of an arithmetic sequence is expressed as

Sn = n/2[2a + (n - 1)d]

Where

n represents the number of terms in the arithmetic sequence.

d represents the common difference of the terms in the arithmetic sequence.

a represents the first term of the arithmetic sequence.

From the information given,

n = 30 terms

a = - 12

d = 7

Therefore, the sum of the first 30 terms, S30 would be

S30 = 30/2[2 × - 12 + (30 - 1)7]

S30 = 15[- 24 + 203)

S30 = 15 × 179 = 2685