Animated Solution for Mathematics - Sequence and Series: If the sum of the first 40 terms of the series, $3+4+8+9+13+14+18+19+\dots$ is $(102)m$, then m is equal to :

The given series is $3+4+8+9+13+14+18+19+\dots$. At first glance, it appears to be a jumble of numbers, but we can identify a heartbeat by looking at the differences between consecutive terms.

The differences are $4-3=1$, $8-4=4$, $9-8=1$, and $13-9=4$. The difference alternates between $1$ and $4$, indicating that this is a structured sequence rather than a standard Arithmetic Progression.

When a series exhibits an alternating difference pattern, the most elegant strategy is to group the terms into pairs. By pairing them, we transform the chaos into order:

Now, we calculate the sums of these pairs: $3+4=7$, $8+9=17$, $13+14=27$, and $18+19=37$. The fog clears, revealing a new series: $7+17+27+37+\dots$.

In our new series, the first term $a$ is $7$, and the common difference $d$ is $17-7=10$. Since we started with $40$ terms and paired them up, our new series has exactly half the number of terms:

We now apply the standard sum formula for an Arithmetic Progression, $S_n=\frac{n}{2}[2a+(n-1)d]$. Substituting our values, we obtain:

We simplify the expression step by step:

The problem states that this sum is equal to $(102)m$. We set up the final equation:

Solving for $m$, we find:

By changing our perspective and grouping the terms, we have successfully solved the problem. The final value is $m=20$.