$\sum_{i=1}^t \phi_i^T(\Lambda_t)^{-1}\phi_i \leq d$ where $\Lambda_t = \lambda I + \sum_{i=1}^{t} \phi_i \phi_i^T$

Question

I came across the following lemma from a paper I am reading

Let $\Lambda_t = \lambda I + \sum_{i=1}^{t} \phi_i \phi_i^T$ where $\phi_i \in \mathbb{R}^d$ and $\lambda > 0 $. Then: \begin{equation} \sum_{i=1}^t \phi_i^T(\Lambda_t)^{-1}\phi_i \leq d \end{equation}

I know how to prove it (using trace and eigen-value decomposition) but I am trying to understand the inequality in an intuitive manner. I have the following questions:

1. How to explain the inequality in a geometric way
2. As $t$ grows how the summation in the inequality behave
3. As $t$ grows the eigenvalues of $\Lambda_t^{-1}$ decreases. How to use this observation to explain the inequality in "words"?
4. What if in the summation we have $\Lambda_i^{-1}$ instead of $\Lambda_t^{-1}$? ie. what can we say about $\sum_{i=1}^t \phi_i^T(\Lambda_i)^{-1}\phi_i$?