sequence and Series (1 Viewer)

[twistedrebel]

If a, b, c forms an arithmetic sequence show that b= a+c/2

 

[annabackwards]

If a, b and c are in an arithmetic series then:
c - b = b - a
c = 2b - a
c + a = 2b
Hence b = (a+c)/2

 

[twistedrebel]

Another two questions if some one could please answer them
1)if a, b, c, d, e are in arithmetic sequence show that a +e = b+d=2c
2)the sum of the first three terms of an arithmetic sequence is 24 and the sum of the next three terms is 51. Find the first term and the common difference of this sequence.

 

[b00m]

a) Prove: a +e = b+d=2c

if a, b, c, d, e are arithmetic sequences:

b - a = c - b = d - c = e - d

---> c-b = d-c [the middle two]
rearrange: 2c = d+b

since

b - a = c - b = d - c = e - d

we know that b-a = e-d

--> b+d = a+e

thus.. 2c = d+b=a+e

 

[bleaver]

2)the sum of the first three terms of an arithmetic sequence is 24 and the sum of the next three terms is 51. Find the first term and the common difference of this sequence.

Ok this method doesn't use the proper formulas, but personally, its easier just to understand what's happening rather than just using a formula.
Let a=First term of your arithmetic sequence.
Let b= The common difference between them.

a+(a+b)+(a+2b)=24
3a+3b=24
a+b=8 --- Equation 1

(a+3b)+(a+4b)+(a+5b)=51
3a+12b=51
a+4b=17 --- Equation 2

Subtracting equation 2 from equation 1:
3b=9
b=3 (common difference)

Substituting back into equation 1:
a=5 (first term)

 

[b00m]

edit: i see someone has beaten me to it.. eh

b) S3 = 3/2 (2a+2d) = 24

and since we know that sum of first 3 is 24.. sum of next 3 is 51, sum of first six is 75

S6 = 3(2a + 5d) = 75

solve simultaneously..

6a+15d = 75
6a + 6d = 48

9d= 27
d=3

Finding first term; put d back into the aove equation(s)

6a + 6(3) = 48
6a=30
a=5

 

[untouchablecuz]

Another two questions if some one could please answer them
1)if a, b, c, d, e are in arithmetic sequence show that a +e = b+d=2c
2)the sum of the first three terms of an arithmetic sequence is 24 and the sum of the next three terms is 51. Find the first term and the common difference of this sequence.