Question

Find the sum of the first 23 terms of the AP 4, −3, −10, . . ..

Answer 1

Answer: -1679

Step-by-step explanation:

In this AP , 7 is subtracted from each number to get the corresponding number .

We have to get the sum of the 1st 23 terms .

So let the first number be x

Then we have x + (x - 7) + (x - 14) ..... + (x - 154)     [154 = 7 * 22]

=> 23x - (7 + 14 + 21 + 28 .... + 154)

=> 23x - (7(1 + 2 + 3 ... + 22)

=> 23x - (7(22 * 23)/2)                            [Formula :- 1 + 2 + 3 + .... + n = $\frac{n(n + 1)}{2}$]

=> 23x - (7*253)

=> 23x - 1771

In this case we have the value of x as 4 .  (Since the sequence 4,-3,-10,... starts with 4 and so on).

So 23*4 - 1771

=> 92 - 1771

=> -1679

So the sum of the first 23 terms of the AP = -1679

Hope this helps you .

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Answer 2

first we have to study about arithemetic progressions and the formulas about this are as follows :

nth term of an AP = an = a ( n - 1)*d

sum of n terms = n/2(2a+n -1)*d

now coming back to the question you have

asked this is for the second formula

given ,

n=23

a =4

d=4 -( -3) =4+3=7

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

=23 ( 2*4 + 23 - 1)*7

_______________

2

=23 ( 8 + 22 )

_____________

2

=23 (30)

________

2

=690÷2

=345

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