An AP consists of 37 terms the sum of 3 middle terms is 225 and sum of last three terms 429 Determine AP.
Answers
SOLUTION:-
Given:
An A.P. consists of 37 terms the sum of 3 middle terms is 225 & sum of last terms is 429.
To find:
The A.P.
Explanation:
We have,
Numbers of terms of A.P. is 37 terms.
We know that, formula of the middle term;
•Three middle most terms will be 18th term, 19th term & 20th term.
Therefore,
18th term= a18
=) a +(18 -1)d
=) a + 17d
19th term = a19
=) a+(19-1)d
=) a + 18d
20th term= a20
=) a+(20-1)d
=) a + 19d
So,
•Sum of all three middle terms= 225
=)(a+17d)+(a+18d)+(a+19d)=225
=) 3a + 54d = 225
=) 3(a +18d)= 225
=) a+18d = 225/3
=) a+18d =75..............(1)
&
•Sum of last three terms = 429
1st last term:
a37
=) a+(37-1)d
=) a +36d
2nd last term:
a37-1
a36
=) a+(36 -1)d
=) a+35d
3rd last term:
a37-2
=) a35
=) a+(35-1)d
=) a +34d
Now,
=) (a+36d)+(a+35d)+(a+34d)= 429
=) 3a + 105d = 429
=) 3(a +35d)= 429
=) a+35d = 429/3
=) a+35d =143...........(2)
Subtracting equation (1) & (2), we get;
=) (a+35d)-(a+18d)= 143-75
=) a-a+35d -18d = 143 -75
=) 17d = 68
=) d= 68/17
=) d= 4
Putting the value of d in equation (1), we get;
=) a +18d = 75
=) a + 18(4) = 75
=) a+ 72 = 75
=) a= 75 -72
=) a=3
So,
The A.P. is 3,7,11,15,19,...
Thank you.
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