The number of terms in an $A.P$. is even; the sum of the odd terms in it is $24$ and that the even terms is $30$. If the last term exceeds the first term by $10 \frac{1}{2},$ then the number of terms in the $A.P$. is :
Let no. of terms $=2n$ $a ,( a + d ),( a +2 d )$, ,........ $a +(2 n -1) d$ sum of even terms $\frac{ n }{2}[2( a + d )+( n -1) 2 d ]=30 \,\,\,\,\, ......(i)$ sum of odd terms $\frac{ n }{2}[2 a +( n -1) 2 d ]=24\,\,\,\,\, .......(ii)$ $a +(2 n -1) d - a =\frac{21}{2}\,\,\,\,\, ......(iii)$ e (i)....e (ii) $\frac{ n }{2} \times 2 d =6$ $ \Rightarrow nd =6 \,\,\,\, ......(iv)$ $(2 n -1) d =\frac{21}{2} \,\,\,\, ......(v)$ $\frac{ eq ( iv )}{ eq ( v )}=\frac{ n }{2 n -1}=\frac{4}{7}$ $ \Rightarrow 8 n -4=7 n $ $n =4$ so no. of terms $=8$
Arithmetic Progression (AP) is a mathematical series in which the difference between any two subsequent numbers is a fixed value.
For example, the natural number sequence 1, 2, 3, 4, 5, 6,... is an AP because the difference between two consecutive terms (say 1 and 2) is equal to one (2 -1). Even when dealing with odd and even numbers, the common difference between two consecutive words will be equal to 2.
In simpler words, an arithmetic progression is a collection of integers where each term is resulted by adding a fixed number to the preceding term apart from the first term.
For eg:- 4,6,8,10,12,14,16
We can notice Arithmetic Progression in our day-to-day lives too, for eg:- the number of days in a week, stacking chairs, etc.
