How to solve: the sum of $t_5$ and $t_{10}$ of a sequence is $10$, which is equal to $t_{16}$. What is $t_{33}$?

Question

This is using arithmetic sequences.

The sum of $t_5$ and $t_{10}$ of a sequence is $10$, which is equal to $t_{16}$. What is $t_{33}$?

($t$ stands for term in the sequence)

Answer 1

Let the first term be a and difference be d .

$t_5+t_{10}=10$

$(a+4d)+(a+9d)=10$ -------- (1)

$t_{16}=10$

$(a+15d)=10$ -------- (2)

Multiply equation (2) by $2$.

$2a+30d=20$ --------- (3)

Subtract equation (1) from (3).

Therefore $d=\frac{10}{17}$.

Put this value in any of the above equation to find $a$.

Then use the formula $t_n= a+(n-1)d$ to find $t_{33}$.

Answer 2

You have $t_n=nd+a$ for some constants $a$ and $d$. You are given $$ t_5+t_{10}=10$$ and $$t_{16}=10 $$ which are two linear equations in the two unknowns $a$ and $d$.