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Sequences and Series — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsSequences and Series1 Mark
Question
State whether statement are True or False.
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
✓

Answer

True.
Solution:
Let a be the first term and d be the common difference of the A.P.
Consider any term ar of an A.P.
Now, $a_{r+m} = ar + (m - 1)d$
And $a_{r-m} = ar + (m - 1)(-d)$
$\therefore a_{r+m} + a_{r-m}= a_r + (m - 1)d + a_r + (m - 1)(-d)$
$\Rightarrow a_{r+m} + a_{r-m} =2a_r$
$\Rightarrow\text{a}_\text{r}=\frac{\text{a}_{\text{r}+\text{m}}+\text{a}_{\text{r}-\text{m}}}{2}$
Thus, any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.

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