Question

In an AP 5,11,17,23…… 419. Prove that the sum of the two middle terms =sum of the first and the last terms

Answer 1

here's your answer!firstly find out no. of term!then dividing it by 2! you'll get no. middle terms!then find thar terms!and do their sum

Answer 2

L=419.
a= 5
d= T2-T1.= 11-5= 6.

First of all we calculate no. of terms . for that.

L= a+(n-1)*d. (L=last term, a=first term, d=common difference of two adjacent terms.)
Now to put values.

419=5+(n-1)*6
414=6n-6
420=6n
n=420/6=70.
Now to find middle two terms we divide no.of terms by 2. Since n is even so we will have two middle terms.
m1= 70/2=35th term and similarly m2= 36th term.
Now we put n=35 the formula .
T= a+(n-1)*d
T35= 5+(35-1)*6= 5+34*6 = 5+204
T35= 209.
Now we put n= 36 in formula
T36 = 5 +(36-1)*6= 5+35*6= 5+210
T36 = 215.
Sum of above two terms is
209+215= 424.
Also sum of last and first terms is
5+419= 424.
Hence sum of middle two terms and sum of last and first terms is equal.