5. If t4 = 22 and t9 = 32, find t8 and S10.

Question

Two terms of an arithmetic progression are given. Find the indicated term and the indicated sum. 

Answer 1

5) Arithmetic series is typically: a, a+d, a+2d, ..., a+(n-1)d, ... where n indicates T_n and d is the common difference, a is the first term.

Therefore, T₄=a+3d=22 and T₉=a+8d=32. T₉-T₄=5d=10, d=2, a=T₄-3d=22-6=16.

Series is 16, 18, 20, 22, 24, 26, 28, 30, 32, ... so T₈=30.

Using the formula a+(n-1)d we also get 16+7×2=30 for T₈.

S₁₀=16+18+20+22+24+26+28+30+32+34=

(16+34)+(18+32)+(20+30)+(22+28)+(24+26)=

5×50=250.

We can also use the formula (n/2)(2a+(n-1)d)=(10/2)(32+18)=250 for S₁₀.