SOLUTION: The 6th term of an AP is 35 and the 13term is 77calculate S6-S4?

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Question 1178050: The 6th term of an AP is 35 and the 13term is 77calculate S6-S4?

Found 3 solutions by MathLover1, chhavi_1997, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
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S%5Bn%5D=S%5B1%5D%2B%28n-1%29%2Ad

if 6th term S%5B6%5D=35 and 13term is S%5B13%5D=77, we have

35=S%5B1%5D%2B%286-1%29%2Ad
35=S%5B1%5D%2B5d.......solve for S%5B1%5D
S%5B1%5D=35-5d.............eq.1

77=S%5B1%5D%2B%2813-1%29%2Ad
77=S%5B1%5D%2B12d.......solve for S%5B1%5D
S%5B1%5D=77-12d.............eq.2

equate right sides of eq.1 and eq.2

35-5d=77-12d.....solve for d
12d-5d=77-35
7d=42
d=6

go to
S%5B1%5D=35-5d.............eq.1, substitute d
S%5B1%5D=35-5%2A6
S%5B1%5D=5

so, your n-th term formula is:

S%5Bn%5D=5%2B%28n-1%29%2A6

now calculate S%5B4%5D
n=4
S%5B4%5D=5%2B%284-1%29%2A6
S%5B4%5D=5%2B3%2A6
S%5B4%5D=23

since given 6th term S%5B6%5D=35

then

S%5B6%5D-S%5B4%5D=35-23=12


Answer by chhavi_1997(1) About Me  (Show Source):
You can put this solution on YOUR website!
Given t6=35
t13=77
therefore, t6= a+(n-1)d=a+(6-1)d
35=a+5d .....1
t13= a+(13-1)d= a+12d
77=a+12d ...2
by solving 1 and 2 we have
a=5, d=6
now, sn=n/2(2a+(n-1)d) using this formula
s6=6/2(2*5+(6-1)*6)
s6=3(10+5*6)
s6=120
similary,s4=4/2(2*5+(4-1)*6)
s4=2(10+3*6)
s4=56
therefore, s6-s4= 120-56
s6-s4=64

Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.


            The problem does not define the meaning of S6, S4, so I will suppose that
            S6 and S4 are the sums of the first 6 and the first 4 terms of the AP, respectively. 


There are 13 - 6 = 7 gaps between the 13-th and 6-th terms of the AP on the number line,

so each gap is  %2877-35%29%2F7 = 6 units.


Thus the common difference of the AP is  d= 6.


Now,  S%5B6%5D - S%5B4%5D = a%5B5%5D + a%5B6%5D = %28a%5B6%5D-d%29 + a%5B6%5D = (35-6) + 35 = 70-6 = 64.


It is the answer to the problem's question:  S%5B6%5D - S%5B4%5D = 64.

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For introductory lessons on arithmetic progressions see
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
    - Finding number of terms of an arithmetic progression
    - Inserting arithmetic means between given numbers
in this site.

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The referred lessons are the part of this online textbook under the topic "Arithmetic progressions".


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