i'm assuming which you have been taught the geometric sequence formula: a million + x + x^2 + ... +x^n = (a million - x^(n+a million)) / (a million - x). This needless to say converges a million/(a million-x) for -a million < x < a million The sequence given on your undertaking is purely a geometrical sequence for x = -r with fee a million/(a million+r)
Comments
Sum of 1 to n = n/2[2a + (n – 1)d
for 1 to 10
sum = 10/2(a + 9d)
5a + 45d = 495
a + 9d = 99 -------- (1)
sum of 1 to 20
= 20/2[2a + 19d]
10a + 190d = 1690
a + 19d = 169 --------(2)
(2) – (1)
10d = 70
d = 7
a + 9d = 99 -------- (1)
a + 63 = 99
a = 36
S30 = 30/2[36 × 2 + 29 × 7]
= 15 × 275 = 4125
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To find the sum of terms in an arithmetic sequence, use the following formula,
Sn = [n/2(a1 + an)]
In this formula:
Sn is the sum of the first n terms in a sequence
n is the number of terms you are adding up
a1 is the first term of the sequence
an is the nth term of the sequence
In this arithmetic series problem:
First 10 terms add upto 495;
that is, 10x + 45 = 495; Hence x = 45
First 20 terms add upto 1090 only
(and not 1690 as given in the problem);
that is, 20x + 190 = 1090; x = 45 (as before)
First 30 terms will add upto 30x + 435
where x = 45.
Hence the sum of the first 30 terms
starting with 45 is 1785.
given
s 10=495
sum 20 =1690
sum of 30 terms=?
sum10= 10/2(a+a10)
495=5a+5a10
sum20=20/2(a+a20)
1690=10a+10a20
a10=a + (10-1)d
a
a20=a + (20-1)d
sum10=5a+5(a+9d)
495=10a+45d ----1
sum20=10a+10(a+19d)
1690=20a+190d ----2
simultaneous 1 and 2
1690-495*2=100d
d=7
a=18
the sum of 30th term is
sum= 30/2(2(18)+29(7))=3585
The first term=18
Difference=7
sum of the first 30 terms=3585
Sn = n/2 (2a + (n-1)d)
so S10 = 5(2a + 9d) = 10a + 45d = 495...eqn 1
and S20 = 10(2a + 19d) = 20a + 190d = 1690...eqn 2
then work them out simulatenously.
by doubling eqn 1 we get:
20a + 90d = 990...eqn 1b
eqn 2 - eqn 1b = 190d - 90d = 700
so 100d = 700
d = 7
sub that into eqn 1 you get
10a + 45(7) = 495
so 10a = 180
so a = 18
subbing back into eqn for Sn
S30 = 15(2(18) + (29)7) = 3585
edit: sorry, made an error in my eqn 1. but i fixed it.
i'm assuming which you have been taught the geometric sequence formula: a million + x + x^2 + ... +x^n = (a million - x^(n+a million)) / (a million - x). This needless to say converges a million/(a million-x) for -a million < x < a million The sequence given on your undertaking is purely a geometrical sequence for x = -r with fee a million/(a million+r)